Structural properties of hierarchically hyperbolic groups

Universidad del Pa´ıs Vasco–Euskal Herriko Unibertsitatea Ph.D. Thesis Structural properties of hierarchically hyperbolic groups. Bruno Robbio Supervised by Ilya V. Kazachkov and Mark F. Hagen Submitted on October 2020 (cc)2020 BRUNO ROBBIO CAMOGLIO (cc by-nc 4.0) 2 Abstract The topics of this dissertation are framed in the area of geometric group theory, that is the study of finitely generated groups through the exploration of its geometric and topological aspects. More precisely, we focus on a class of groups called hierarchically hyperbolic groups. Hierarchical hyperbolicity is a very recent but powerful notion whose goal is to provide a unifying framework to study large classes of groups having features reminiscent of non-positive and negative curvature. We include an introduction to this class of groups in the first chapter. The first original results of this thesis appear in Chapter 2, where a number of structural results on hierarchically hyperbolic spaces are proved. In addition, two notions are presented here: the intersection property and concreteness. These key conditions are used in numerous places throughout the rest of the thesis and are crucial for understanding the main results that follow. The first main contribution of the thesis is the establishing of a combination theorem for the class of hierarchically hyperbolic groups. We usually refer to a result as a combination theorem on a class of groups Cif it provides an answer to the following question: Let Gbe a group acting on a simplicial tree Twith vertex and edge stabilizers in C, under what conditions can we conclude that the group Gis itself in C? In our case, the conditions that we identified are the intersection property and clean containers. As an application of this theorem we obtain that graph products of hierarchically hyperbolic groups with the intersection property and clean containers are themselves hierarchically hyperbolic. In the last chapter of the thesis we focus on the class of groups that act on a simplicial tree such that the vertex stabilizers are hyperbolic and edge stabilizers are virtually cyclic. We call this class hyperbolic-2-decomposable groups. We obtain a characterization of groups of this type that allows us to provide a hierarchical hyperbolic structure on them. More precisely, we obtain that a hyperbolic-2-decomposable group is hierarchically hyperbolic if and only if it is balanced. Even more, we show that this is equivalent to the group itself not containing non-euclidean BaumslagSolitar subgroups. As an immediate corollary we obtain that free products with amalgamation of hyperbolic groups over virtually cyclic groups are hierarchically hyperbolic. i ii Resumen Los temas de esta tesis se enmarcan en el ´area de la teor´ıa geom´etrica de grupos, que es el estudio de grupos finitamente generados a trav´es de la exploraci´on de sus aspectos geom´etricos y topol´ogicos. M´as precisamente, nos centramos en una clase de grupos denominados grupos jer´arquicamente hiperb´olicos. La hiperbolicidad jer´arquica es una noci´on muy reciente pero poderosa cuyo objetivo es proporcionar un marco unificador para estudiar grandes clases de grupos que tienen caracter´ısticas similares a curvatura negativa y no positiva. Inclu´ımos una introducci´on a ´esta clase de grupos en el primer cap´ıtulo. Los primeros resultados originales de esta tesis aparecen en el cap´ıtulo 2, donde se prueban una serie de resultados estructurales sobre espacios jer´rquicamente hiperb´olicos. Se presentan, adem´as, dos nociones: intersection property y concreteness. Estas condiciones se utilizan en varios lugares a lo largo del resto de la tesis y son cruciales para comprender los principales resultados que siguen. La primera contribucin principal de la tesis es el establecimiento de un teorema de combinaci´on para la clase de grupos jer´arquicamente hiperb´olicos. Por lo general, nos referimos a un resultado como un teorema de combinaci´on en una clase de grupos Csi responde a la siguiente pregunta: Sea Gun grupo que act´ua sobre un ´arbol simplicial Tcuyos estabilizadores de v´ertices y aristas pertenecen a C, bajo qu´e condiciones podemos concluir que el grupo Gest´a en C? En nuestro caso, las condiciones que identificamos son intersection property y clean containers. Como aplicaci´on de este teorema obtenemos que los productos bajo grafos de grupos jer´arquicamente hiperb´olicos con intersection property y clean containers son en s´ı mismos jer´arquicamente hiperb´olicos. En el ´ultimo cap´ıtulo de la tesis nos centramos en la clase de grupos que act´uan sobre un ´arbol simplicial de manera que los estabilizadores de aristas son virtualmente c´ıclicos. Llamamos a esta clase grupos hyperbolic-2-decomposable. El principal resultado de ´este ´ultimo cap´ıtulo es una caracterizaci´on de grupos de este tipo que nos permiten aportar una estructura hiperb´olica jer´arquica sobre ellos. M´as precisamente, obtenemos que un grupo hyperbolic-2-decomposable es jer´arquicamente hiperb´olico si y solo si es equilibrado. A´un m´as, mostramos que esto es equivalente a que el grupo en s´ı no contenga subgrupos de tipo Baumslag-Solitar no equilibrados. Como corolario inmediato obtenemos que los productos libres amalgamados de grupos hiperb´olicos sobre grupos virtualmente c´ıclicos son jer´arquicamente hiperb´olicos. Contents i Introduction v 1 Preliminaries 1 1.1 Geometryofgroups ................................... 1 1.2 Hyperbolicgroups .................................... 3 1.2.1 Quasiconvexity.................................. 4 1.3 Graph of groups and Bass-Serre Theory . . . . . . . . . . . . . . . . . . . . . . . . 5 1.4 Relatively hyperbolic groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 1.5 Hierarchically hyperbolic spaces: introduction . . . . . . . . . . . . . . . . . . . . . 11 1.5.1 Projections and coordinate system . . . . . . . . . . . . . . . . . . . . . . . 12 1.5.2 Constructing structures in main examples . . . . . . . . . . . . . . . . . . . 12 1.6 Hierarchically hyperbolic spaces: full definition . . . . . . . . . . . . . . . . . . . . 14 1.7 Hierarchically hyperbolic groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 1.7.1 Hierarchical quasiconvexity and gate maps . . . . . . . . . . . . . . . . . . . 18 1.7.2 Gatemaps .................................... 20 1.8 Productregions...................................... 20 1.9 Constructing examples of hierarchical hyperbolicity . . . . . . . . . . . . . . . . . . 23 1.9.1 Hierarchically hyperbolic structures on groups acting on trees . . . . . . . . 25 1.9.2 A characterization of hierarchical hyperbolicity in hyperbolic-2-decomposable groups....................................... 28 1.9.3 Anoteontorsion................................. 30 2 Structural results 33 2.1 Intersection property and concreteness . . . . . . . . . . . . . . . . . . . . . . . . . 34 2.2 ProofofthemainTheorem ............................... 43 2.3 Mainstructuralresults.................................. 49 3 A Combination theorem 53 3.1 Trees of hierarchically hyperbolic spaces . . . . . . . . . . . . . . . . . . . . . . . . 54 iii iv CONTENTS 3.1.1 Trees with decorations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57 3.2 Endowing a tree of HHS with an HHS structure . . . . . . . . . . . . . . . . . . . . 60 3.2.1 Construction of index set . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60 3.2.2 Hyperbolic spaces associated to the index set and projections . . . . . . . . 63 3.2.3 Projections between hyperbolic spaces . . . . . . . . . . . . . . . . . . . . . 64 3.2.4 Proof of the main theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67 3.3 Applications........................................ 78 3.3.1 Graph of hierarchically hyperbolic groups . . . . . . . . . . . . . . . . . . . 79 3.3.2 Graphproducts.................................. 86 4 Hyperbolic-2-decomposable groups that are HHG 91 4.0.1 Questions..................................... 91 4.0.2 Balancedgroups ................................. 92 4.0.3 Convexity..................................... 96 4.1 Hierarchical hyperbolicity of (2-ended)-2-decomposable groups . . . . . . . . . . . 97 4.1.1 Two-endedgroups ................................ 97 4.1.2 Pulling back hierarchical structures . . . . . . . . . . . . . . . . . . . . . . . 98 4.1.3 Linearly parametrizable graph of groups . . . . . . . . . . . . . . . . . . . . 99 4.1.4 Characterizations of hierarchical hyperbolicity . . . . . . . . . . . . . . . . 103 4.2 Hierarchical hyperbolicity of hyperbolic-2-decomposable groups . . . . . . . . . . . 106 4.2.1 Commensurability and conjugacy graph . . . . . . . . . . . . . . . . . . . . 109 Bibliography 117 Introduction Hierarchically hyperbolic spaces and groups (HHSs and HHGs) were introduced by Behrstock, Hagen and Sisto in a series of papers [12, 14]. This is a broad class that includes an impressive amount of spaces and groups naturally occurring from geometric considerations. Mapping class group of surfaces; CAT(0)-cube complexes; Teichmuller space with the Teichmuller and WeilPetersson metric and fundamental groups of 3-manifolds with no Nil nor Sol component are among the most famous objects admitting a hierarchical hyperbolic structure. Several generalizations of hyperbolic groups have been introduced over the years to describe groups of geometric origin that exhibit some notion of negative curvature. Relative hyperbolicity ([22, 37]) recovers fundamental groups of 3-manifolds with cusps, whereas mapping class groups are examples of acylindrically hyperbolic groups [69], and raags (that is right-angled Artin groups) are among the groups acting properly and cocompactly on CAT(0) cube complexes, that is cubulable groups [76, 91]. Moreover, mapping class groups are not relatively hyperbolic (unless they are already hyperbolic [8, Theorem 1.2]). The notion of hierarchical hyperbolicity emerges as a class that generalizes hyperbolicity, engulfs many of the above mentioned groups and also maintains many of their algebraic features. Being a hierarchically hyperbolic group presents a wide range of both algebraic and geometric consequences. Some of these are a quadratic isoperimetric inequality; finite asymptotic dimension; a version of the Tits alternative; rank ridigity theorems and a controlled way in which quasi-flats are distributed in the group. The key insight to define hierarchically hyperbolic groups is the axiomatization of the MasurMinsky machinery developed for mapping class groups for general groups. Efforts in this direction are not a novelty in certain classes of groups. For instance, in [80] the author presents a way of characterizing relative hyperbolicity in terms of projections similar to that of subsurface projections in the curve graph and develops a distance formula. Moreover, in [46], the author introduces the contact graph for cubical groups, an analog of the contact graph for cube complexes. A hierarchical hyperbolic structure on a geodesic metric space Xis composed of the following data: 1. An index set S; 2. a collection of δ-hyperbolic spaces; 3. a collection of projections tπV:XÑCVuVPS. This data must satisfy a set of axioms. The full definition is included in Section 1.6. v vi INTRODUCTION Organization of the thesis This thesis is divided into four chapters. Chapter 1 is expository and recollects basic concepts on coarse geometry and geometry of groups. The main topics included in this chapter are hyperbolic groups (Section 1.2), Bass-Serre theory (Section 1.3), and relatively hyperbolic groups (Section 1.4). We also give an introduction to the definition of hierarchically hyperbolic spaces and groups (Section 1.6), which are the main object of study throughout the rest of the work. The final section of this first chapter (Section 1.9) deals with examples of hierarchically hyperbolic groups, and is intended as an introduction and motivation for the original work that is presented in this thesis. The reader that is well-versed in hierarchical hyperbolicity may wish to start the reading of the thesis in this section. The remaining chapters comprise the original contributions of the author, with Chapters 2 and 3 being part of a joint work with Federico Berlai ([15]) and Chapter 4 part of a joint work with Davide Spriano ([71]). Chapter 2 concentrates on structural properties of hierarchically hyperbolic spaces and hieromorphisms (i.e morphisms in the class of HHGs). We introduce the notions of intersection property, of ε-support, and of concreteness of a hierarchically hyperbolic space (see Definition 2.1.1, Definition 2.1.6, and Definition 2.1.10). All of these will be necessary for Chapter 3. The main theorem of this chapter is Theorem 2.2.1 which is then used in the proofs of Theorem 2.3.3 and Lemma 2.3.4. These results will be applied repeatedly in Chapter 3, which is devoted to the proof of Theorem 3.0.1. Chapter 3 we present and prove a combination theorem on hierarchically hyperbolic spaces (Theorem 3.0.1). Section 3.1 is concerned with trees of hierarchically hyperbolic spaces, which is an extension of the notion of trees of spaces to the class of HHSs. In Subsection 3.1.1 we introduce a trick, which we call the decoration of a tree of hierarchically hyperbolic spaces T, which is fundamental for our approach to prove Theorem 3.0.1. To a tree of HHSs Twe associate a total space XpTqthat, in Section 3.2, we prove that can be endowed with a hierarchical hyperbolic structure. Section 3.3 is concerned with two applications of Theorem 3.0.1. The first one, Corollary 3.3.1 is a combination theorem for hierarchically hyperbolic groups. As a byproduct of Theorem 3.3.7, we extend the results of [2] to show that clean containers are not only preserved by taking free and direct products, but also by graph products. Chapter 4 is devoted to the application of the combination theorem developed in the previous chapter to groups that split as graphs of groups with hyperbolic vertex groups and 2-ended edge subgroups. To abbreviate, if Pis a property of a group, we say that a group is P-2-decomposable if it splits as a graph of groups with 2-ended edge groups and vertex groups satisfying property P. The main result of this chapter is that a hyperbolic-2-decomposable group has a hierarchical hyperbolic structure if and only if it is balanced (Corollary 4.2.16). If the group is further assumed to be virtually torsion-free, we obtain that a hyperbolic-2-decomposable group is hierarchically hyperbolic if and only if contains no non-euclidean Baumslag-Solitar subgroup (Corollary4.2.15). In Section 4.1 we introduce the notion of linear parametrization (Definition 4.1.11) on (2-ended)-2decomposable groups and use this to prove the main result of this chapter for that class (Theorems 4.1.25 and 4.1.24). In Section 4.2 we prove Theorem 4.2.2, which allows us to extend the results developed in Section 4.1.4 to the more general class of hyperbolic-2-decomposable groups. Chapter 1 Preliminaries This chapter is meant as an introduction to the main aspects of geometric group theory, aimed at presenting hierarchically hyperbolic spaces and groups and how they fit into the area. We begin by recalling the basic definitions and objects that will appear throughout the section. 1.1 Geometry of groups Definition 1.1.1. Let Gbe a group and let Xbe a metric space such that Gacts on Xby isometries. We say that the action is 1. properly discontinuous if for all compact KĎX, |tgPG|gK XK‰∅u| ă 8. 2. cocompact if X{Gis compact in the quotient topology. 3. The metric space Xis proper if closed balls are compact. We often use the abbreviation of geometric action to refer to a properly discontinuous and cocompact action of Gon X. From now on, when we say that a group Gacts on a metric space Xwe assume that the action is by isometries, unless otherwise stated. Definition 1.1.2 (Cayley graph). If Gis a group generated by a finite set S“ ts1, . . . , snuwe associate a graph Xto the pair pG, Sqwhere the underlying vertex set is Gand two elements g, h are at distance one in Xif and only if g´1hbelongs in S. This graph Xis known as the Cayley graph of Gwith respect to S. Associating a Cayley graph to a finitely generated group can be viewed as a process that converts groups to metric spaces. Further, it is straightforward to check that a finitely generated group acts geometrically on any of its Cayley graphs. We use X“CaypG, Sqto denote the Cayley graph of a group with respect to a generating set S. A crucial observation says that the large-scale structure of a Cayley graph does not depend on the choice of generating set. This observation is usually referred to as the Milnor-Svarc lemma: 1 8CHAPTER 1. PRELIMINARIES the subword must appear in “ptf11tf2. . . tfsq`u0t1 e1u1. . . tm emum˘‰. Since uwas assumed to be reduced and f1, . . . , fsis a shortest path, the subword must be tfsu0t1 e1, where u0“φf` spzqfor some zPGfs. Then replace tfsu0t1 e1by φf´ spzq, and perform the symmetric change on the other side of the x. Note that this process reduces the length of the path f1, . . . , fsby one. In particular, it has to terminate. So, assume that no reduction can be performed in pu ““ptf11tf2. . . tfsq`u0t1 e1u1. . . tm emum˘‰. If pu “h0PGw, and hence x, y PGware conjugate in Gw, we are done. So suppose this is not the case. We need to have that umxu´1 m“φ` empzqfor some zPGem. Substitute tm emumxu´1 mt´m emwith tm´1 emφe´ mpzqt´m`1 em. If mą1, add a path contained in the spanning tree and repeat the process using the normal form theorem again, until we obtain a reduction of the form tm´1 em´1um´1Z0u´1 m´1t´m´1 em´1, for some Z0Pφe´ mpGemq. Again, we must have um´1Z0u´1 m´1Pφe` m´1pGem´1, that is to say, um´1φe´ mpGemqu´1 m´1Xφem´1pGem´1q‰t1u. Proceeding as above, we get the claim for each ui. Whenever we are working on a graph of groups, it is often the case that we are interested in studying a subgraph of groups. For that we adopt the following notation. Notation. Let Gbe a graph of groups and Γ its underlying graph. If Λ ĎΓ is a connected subgraph, then we can define the subgraph of groups G|Λ, where the underlying graph is Λ, every vertex and edge in Λ has the same associated groups as in Gand the maximal subtree of Γ is an extension of the maximal subtree of Λ. We call G|Λthe subgraph of groups spanned by Λ. Lemma 1.3.11. Let Gbe a graph of groups and let ΛĎΓbe a subgraph. Let T1ĎΛbe a spanning tree of Λsuch that T1can be extended to the spanning tree Tin Γ. Then, there exists a group injection π1pG|Λ, T 1qãÑπ1pG, Tq. Remark 1.3.12. 1. If Γ consists of a single vertex vand a single edge e, then π1pGqis isomorphic to the HNN extension Gv˚φe. 2. If Γ consists of two vertices v, w and a single edge ejoining them, then π1pGqis isomorphic to the free product with amalgamation Gv˚GeGw. 3. Whenever Γ is a tree, we will call π1pGqatree product. Definition 1.3.13. We say that a group Gsplits non-trivially if there exists a graph of groups G such that G–π1pGqand such that Gis not isomorphic to Gvor Gefor any vPVpΓqand ePEpΓq. We now recall the fundamental theorem relating splittings of a group with groups acting on trees. This is also known as the fundamental Bass-Serre theorem. 1.4. RELATIVELY HYPERBOLIC GROUPS 9 Theorem 1.3.14. Let Gbe a group that splits non-trivially as G–π1pGq. Then, there exists a tree Ton which Gacts without edge inversion such that the factor graph T{Gis equal to ΓG. Moreover, the stabilizers of vertices and edges of this action are conjugate to vertex and edge groups in Grespectively. Proof. See, for instance, [19, Theorem 12.1] and [19, Theorem 15.1]. Note that the tree Tdepends on the splitting of the group G. Conversely, the splitting of a group Gis determined in terms of both the tree on which Gacts and the action. Definition 1.3.15. We call the tree Tassociated to a splitting Gof Gthe Bass-Serre tree. 1.4 Relatively hyperbolic groups As already stressed by Gromov, some natural groups of geometric origin do not fit into the hyperbolicity picture: Kleinian groups and fundamental groups of 3-manifolds with cusps are examples of this fact. He also noticed that even spaces which are not hyperbolic may present some hyperbolic-like features in its geometry. More precisely, in [44], he describes a family of spaces where the absence of hyperbolicity is restricted to an isolated finite collection of subgroups. In a group theoretical language, these are groups where the Cayley graph is hyperbolic outside of a finite collection of subgroups. These groups are known as relatively hyperbolic groups, a class that generalizes hyperbolic groups. Relative hyperbolicity was formally introduced independently by B. Farb and B. Bowditch in [22,37]. Ever since, relatively hyperbolic groups has been extensively studied and shown to be an extremely rich object to analyse from multiple points of view. To name a few, relatively hyperbolic groups have been studied in relation with algorithmic properties ([68]); asymptotic cones ([34]); and quasi-flats ([27]). Moreover, a characterization of relative hyperbolicity in terms of projections has been developed in [80]. There are multiple definitions of relatively hyperbolic groups in the literature (see, for instance [22,37,44]). In this chapter we include the one due to Bowditch in [22]. Definition 1.4.1. Let Gbe a finitely generated group and let H1, H2, . . . , Hkbe a subgroup of G. We say that Gis hyperbolic relative to H1, . . . , Hkif Gacts on a hyperbolic graph Xwith the following conditions: 1. The number of orbits of edges is finite; 2. finite edge stabilizers; 3. vertex stabilizers are either finite or conjugate to some Hi; 4. the graph Xis fine: for every nPNand any edge of Xis contained in finitely many circuits of length n. Here, by circuit we mean a cycle without self-intersection). 10 CHAPTER 1. PRELIMINARIES Figure 1.1: Coned-off Cayley graph of Gwith respect to H. We call a peripheral subgroup to each one of the subgroups Hi. Examples/Properties 1.4.2. 1. If H1, H2are hyperbolic groups and Fis a common finite subgroup then G“H1˚FH2is hyperbolic relative to tH1, H2u. Indeed, the action of Gon Xthe Bass-Serre tree corresponding to H1˚FH2satisfies the conditions of Definition 1.4.1; 2. The group Z2“ xa, b | ra, bsy is weakly hyperbolic with respect to xaybut it is not hyperbolic relative to it. Indeed, if that were the case, then xaywould stabilize a vertex vin Xand xayb would stabilize a vertex wjoined by an edge to v. Thus, xay X xayb“ xaywould stabilize that edge. This contradicts condition 2 of Definition 1.4.1 3. If Gis hyperbolic relative to a subgroup Hthen His almost malnormal in G(i.e |HgXH|ă8 for every gPGzH). 4. Let Gbe hyperbolic relative to a subgroup HďG. If His hyperbolic, then Gis hyperbolic. A useful construction when studying relative hyperbolicity is the coning-off of a group with respect to a collection of subgroups. This will be particularly helpful when we consider the relative hyperbolicity in terms of the associated Cayley graph itself instead of an abstract graph. Definition 1.4.3. [Coned-off Cayley graph] Let Gbe a finitely generated group and let Hbe a finitely generated subgroup of G. Fix a set of generators Sof G. In the Cayley graph CaypG, Sq add a vertex vpgHqfor each left coset gH of H, and connect vpgHqwith each xPgH by an edge of length 1{2. The obtained graph y CaypG, Sqis called a coned-off graph of Gwith respect to H. We give this graph the path metric. We say that Gis weakly hyperbolic relative to Hif y CaypG, Sqis aδ-hyperbolic metric space for some δas in Definition 1.2.2. Note that y CaypG, Sqis not a proper metric space, as closed balls are not necessarily compact. 1.5. HIERARCHICALLY HYPERBOLIC SPACES: INTRODUCTION 11 Remark 1.4.4. It is easy to see that y CaypG, Sqis quasi-isometric to the graph obtained from CaypG, Sqby collapsing each left coset of Hto a point. However, the coned-off Cayley graph y CaypG, Sqwith respect to His quite different from the graph CaypG, Sq{Hobtained from quotienting the action of Hon CaypG, Sq. This is due to the difference between left and right cosets of Hin G. If His normal in Gthen y CaypG, Sqand CaypG, Sq{Hare quasi-isometric. Lemma 1.4.5. If Gis hyperbolic relative to a collection Pthen the hyperbolic graph Xcan be taken to be the coned-off Cayley graph of Gwith respect to P. Lastly, we include two results on relative hyperbolic groups that anticipate much of the following chapter. Lemma 1.4.6. [Projections][34, Lemma 4.11] Let Gbe a group hyperbolic relative to a finite collection of subgroups tH1, . . . , Hku. If Pis the set of left cosets of peripherals in G. For each PPPthe closest-point projection πP:GÑPis a coarsely Lipschitz map. Theorem 1.4.7. There exists s0so that for every sěs0there exists K, C so that for every x, y PG dpx, yq —pK,Cqÿ PPPttdpπPpxq, πPpyqquus`dp Gpx, yq. 1.5 Hierarchically hyperbolic spaces: introduction Despite its success, relative hyperbolic groups are far from completing the picture of groups with hyperbolic-like features. Perhaps the most well-known evidence of this fact are Mapping class groups of surfaces. Indeed, it has been shown in [6, 8] that mapping class group of a surface of complexity at least one can never be hyperbolic relative to any collection of finitely generated subgroups. However, the powerful Masur-Minsky machinery ([63,64]) developed for these groups is a clear indicative of the manifestation of hyperbolicity in it. Therefore, one is brought to find a set of properties that would generalize hyperbolicity, include mapping class groups, and still have strong algebraic consequences for groups satisfying them. These conditions have been identified by Behrstock, Hagen, and Sisto, who isolated the notions of hierarchically hyperbolic spaces and of hierarchically hyperbolic groups [12,14]. Again, the geometric approach that is undertaken reflects into strong algebraic and asymptotic properties: hierarchically hyperbolic groups are finitely presented [14, Corollary 7.5], they satisfy a quadratic isoperimetric inequality [14, Corollary 7.5], they are coarse median [14, Theorem 7.3], and they have finite asymptotic dimension [10]. The definition of hierarchically hyperbolic spaces is quite technical and lengthy. Thus, before we present the full definition we would like to devote some space to properly motivate and introduce every significant aspect of this class. The emphasis of this section is put on a heuristic approach to the construction of hierarchical hyperbolic structures rather than a technical overview of the theory. The experienced reader may wish to skip this section. 12 CHAPTER 1. PRELIMINARIES 1.5.1 Projections and coordinate system A hierarchical hyperbolic structure on a geodesic metric space Xconsists of the following data: 1. A collection of δ-hyperbolic spaces tCVu; 2. a set Sthat indexes the various hyperbolic spaces; 3. for every VPS, a pK, Kq-coarsely Lipschitz map πV:XÑCV. The set of indices along with the various hyperbolic spaces endow Xwith a coordinate system that allows to investigate the geometric aspects of Xby means of its projections. Following this spirit, a hierarchically hyperbolic space can be roughly thought of as a metric space that can be decomposed into building blocks that are hyperbolic metric spaces. The most basic example of a space with this characteristics is R2, as it can clearly be decomposed as a direct product of two infinite lines. The defining structure of a hierarchically hyperbolic space also contains three relations that encode how do various elements in the index set relate to each other. These are called nesting (denoted by Ď); transversality (denoted by &) and orthogonality (denoted by K). Each one of this relations impose conditions in which the way the hyperbolic building blocks fit in X. 1.5.2 Constructing structures in main examples Here we describe the hierarchical hyperbolic structure in different classes of groups. Right-angled Artin groups Let Γ be a simplicial graph. We recall that the Right-angled Artin group associated to Γ is defined as the group given by the presentation AΓ“ xVpΓq|rv, ws “ 1ô tv, wu P EpΓqy. The space CaypAΓqcan be endowed with a hierarchically hyperbolic structure as follows. (Index set) Let PΓbe the collection of all full subgraphs of Γ. For each Λ PPΓwe say that two cosets gAΛ, hAΛare parallel if rgh´1, AΛs “ 1. Note that parallelism defines an equivalence relation on the set of cosets of tAΛ|ΛPPΓu. We use rgΛsto denote the parallelism class of the coset gAΛfor each Λ PPΓ. We set the index set Sto be trgΛs | ΛPPΓ, g PAΓu. (Hyperbolic spaces) To each rgΛs P Swe associate the hyperbolic space CrgΛsdefined as gp AΛ, where p AΛis the Cayley graph of AΛwith SΛ“VpΓqYtAΛ1ăAΛ|Λ1ĹΛuas generating set. Theorem 1.5.1. [12] The space CrgAΛs “ gp AΛis quasi-isometric to a tree, in particular it is hyperbolic. (Projections) For each rgΛs P Swe associate the projection πΛ:AΓÑCrgΛsas the composition ι˝pΛ. Here, pΛdenotes the closest-point projection onto gAΛin the Cayley graph of AΓwith the standard generating set and ιis the inclusion CaypAΛ;VpΛqq Ñ CaypAΛ;SΛq. 1.5. HIERARCHICALLY HYPERBOLIC SPACES: INTRODUCTION 13 Graph of multicurves We would now like to outline the hierarchical hyperbolic structure on a graph of multicurve. Let us first recall some notions. Let S“Sg,n denote the connected, oriented surface of genus gwith npunctures. The complex of curves CSassociated to Swas originally introduced by Harvey [50]. It is defined as a complex where the 1-skeleton is given by the following: 1. Vertices: There is one vertex for each isotopy class of essential simple closed curve in S. 2. Edges: There is an edge between pair of vertices in CSwhenever the corresponding isotopy class of curves can be realized disjointly. We assume that every edge in CShas length one, making it a metric space. This means that if α, β are curves in Ssuch that dCSprαs,rβsq “ nthen there exist curves α“α1, . . . , αn“βsuch that rαisand rαi`1scan be realized disjointly for every i. While the mapping class group of surfaces are almost never hyperbolic, the following groundbreaking result by Masur and Minsky evidences a connection between the mapping class group of a surface and negative curvature. Theorem 1.5.2. [63] There exists δsuch that CSis δ-hyperbolic, where δdepends on S. We now recall an important tool developed by Masur and Minsky. For any subsurface S1of S we define the subsurface projection map πS1:CSÑ2CS1as follows. Let αbe a curve realized in minimal position with BSS1( that is to say, the number of points in the intersection αXBS1is minimal in terms of isotopy). If αis contained in S1, we define πS1pαqas α. If αis disjoint from S1, we define πS1pαqas ∅. Otherwise, for each arc ωof intersection of αwith S1, we take the boundary component of a small regular neighbourhood of ωYBSS1which are non-peripheral in S1. Then, we set πS1pαqas the union of these curves over all such ω. The above projection system can be extended to various types of so-called graphs of multicurves in S. A graph of multicurves is defined as a graph associated to a surface where each vertex corresponds to a collection of isotopy class of curves in S. This notion extends the one of curve graph of a surface and, over the past decades it has attracted significant attention. In [88] the author shows that a wide range of examples of curves of this type are hierarchical hyperbolic. We now focus on the hierarchically hyperbolic structure on a specific graph of multicurves called the pants decomposition graph, which we denote by GpSq. Each vertex in GpSqcorresponds to a multicurve on Sthat defines a pant decomposition. Two vertices v, w in the pants decomposition graph are joined by an edge if one of the curves αvin vcan be replaced by a curve αwin w. (Index set) We define the index set Sas the collection of isotopy classes of all possible subsurfaces of S. (Hyperbolic spaces) We associate to every S1PSthe curve graph CS1. (Projections) For each SPSwe associate the map πS:GpSq Ñ CSdefined as the subsurface projection described above. For an explicit proof of the hierarchical hyperbolicity of many graphs of multicurves we refer to [88]. 14 CHAPTER 1. PRELIMINARIES The two examples above illustrate one of the most remarkable features in the theory of hierarchically hyperbolic spaces: it is a class of spaces that engulfs various objects that seem to be inherently different from a geometric viewpoint. For instance, a Cayley graph of a right-angled Artin group is an example of a CAT(0)-cube complex, whereas the mapping class group of a surface is almost never a CAT(0) metric space ([25,55])). Groups hyperbolic relative to hierarchically hyperbolic groups If Gis a group which is hyperbolic relative to a collection of hierarchically hyperbolic groups tpHi,SHiqun i“1then CaypGqcan be endowed with a hierarchically hyperbolic group structure. For each i“1...,n and each left coset of Hiin G, fix a representative gHi. Let gSibe a copy of Si. Let p Gbe the hyperbolic space obtained by coning-off Gwith respect to the peripherals tHiu. (Index set) We define the index set as S“ tp GuYŮgPgŮiSgHi. (Hyperbolic spaces) For each copy gV of an element VPSHiwe associate a copy of CVas CgV . (Projections) πp G:GÑp Gis the inclusion, which is coarsely surjective and hence has quasiconvex image. For each UPSgHi, let ggHi:GÑgHibe the closest-point projection onto gHiand let πG U“πHi U˝ggHi, to extend the domain of πUfrom gHito G. Since each πHi Uwas coarsely Lipschitz on CUwith quasiconvex image, and the closest-point projection in Gis uniformly coarsely Lipschitz (Lemma 1.4.6), the projection πG Uis uniformly coarsely Lipschitz and has quasiconvex image. 1.6 Hierarchically hyperbolic spaces: full definition The definition of Hierarchically hyperbolic spaces and groups can be found in [12] and [14]. It is also worth mentioning that in [82] a very accessible and friendly introduction can be found. We now present the definition of hierarchically hyperbolic spaces and groups in its full generality and subsequently examine the various ingredients in detail. Definition 1.6.1. Aq-quasigeodesic metric space pX, dXqis hierarchically hyperbolic if there exist δě0, an index set S, and a set tCW|WPSuof δ-hyperbolic spaces pCU, dUq, such that the following conditions are satisfied: 1. (Projections) There is a set tπW:XÑ2CW|WPSuof projections that send points in X to sets of diameter bounded by some ξě0 in the hyperbolic spaces CWPS. Moreover, there exists Kso that all WPS, the coarse map πWis pK, Kq-coarsely lipschitz and πWpXq1is K-quasiconvex in CW. 2. (Nesting) The index set Sis equipped with a partial order Ďcalled nesting, and either S is empty or it contains a unique Ď-maximal element. When VĎW,Vis nested into W. For each WPS,WĎW, and with SWwe denote the set of all VPSthat are nested in W. For all V, W PSsuch that VĹWthere is a subset ρV WĎCWwith diameter at most ξ, and a map ρW V:CWÑ2CV. 1If AĎX, by πUpAqwe mean ŤaPAπUpaq. 1.6. HIERARCHICALLY HYPERBOLIC SPACES: FULL DEFINITION 15 3. (Orthogonality) The set Shas a symmetric and antireflexive relation Kcalled orthogonality. Whenever VĎWand WKU, then VKUas well. For each ZPSand each UPSZ for which tVPSZ|VKUu‰H, there exists contZ KUPSZztZusuch that whenever VKU and VĎZ, then VĎcontZ KU. 4. (Transversality and Consistency) If V, W PSare not orthogonal and neither is nested into the other, then they are transverse: V&W. There exists κ0ě0 such that if V&W, then there are sets ρV WĎCWand ρW VĎCV, each of diameter at most ξ, satisfying min dWpπWpxq, ρV Wq, dVpπVpxq, ρW Vq(ďκ0,@xPX. Moreover, for VĎWand for all xPXwe have that min dWpπWpxq, ρV Wq,diamCVpπVpxqYρW VpπWpxqqq(ďκ0. In the case of VĎW, we have that dUpρV U, ρW Uq ď κ0whenever UPSis such that either WĹU, or W&Uand UMV. 5. (Finite complexity) There is a natural number ně0, the complexity of Xwith respect to S, such that any set of pairwise Ď-comparable elements of Shas cardinality at most n. 6. (Large links) There exist λě1 and Eěmaxtξ, κ0usuch that, given any WPSand x, x1PX, there exists tTiui“1,...,tNuĂSWztWusuch that for all TPSWztWueither TPSTi for some i, or dTpπTpxq, πTpx1qq ă E, where N“λdWpπWpxq, πWpx1qq ` λ. Moreover, dWpπWpxq, ρTi Wq ď Nfor all i. 7. (Bounded geodesic image) For all WPS, all VPSWztWuand all geodesics γof CW, either diamCVpρW Vpγqq ď Eor γXNEpρV Wq‰H. 8. (Partial realization) There is a constant αsatisfying: let tVjube a family of pairwise orthogonal elements of S, ad let pjPπVjpXq Ď CVj. Then there exists xPXsuch that •dVj`πVjpxq, pj˘ďαfor all j; •for all jand all VPSsuch that V&Vjor VjĎVwe have dVpπVpxq, ρVj Vq ď α. 9. (Uniqueness) For each κě0 there exists θu“θupκqsuch that if x, y PXand dpx, yq ě θu, then there exists VPSsuch that dVpx, yq ě κ. The inequalities of the fourth axiom are called consistency inequalities. Remark 1.6.2. The element contZ KUappearing in Axiom (3) of Definition 1.6.1 is called the orthogonal container (or the container of the orthogonal complement) of Uin Z. If Zis the 16 CHAPTER 1. PRELIMINARIES Ď-maximal element of S, then we might suppress it from the notation, write contKUand call it higher container. If Zis not the Ď-maximal, then we will talk about lower containers. A hierarchically hyperbolic space has clean containers if UKcontZ KUfor all U, Z PS, as originally defined in [2, Definition 3.4]. For a hierarchically hyperbolic space pX,Sqand a subset UĎS, we define (1.1) UK:“ tVPS|VKUfor every UPUu. We usually use the tuple pX,Sqto denote a hierarchically hyperbolic space, where Xis a metric space and Sis the collection of δ-hyperbolic spaces. Before diving deeper into the theory, let us show a few basic examples of hierarchically hyperbolic spaces. Examples/Properties 1.6.3. 1. If Xis hyperbolic, then pX,tXuq is a hierarchically hyperbolic space structure, where the projection πXis idX; 2. If Z2“ xa, b | ra, bsy then we can endow Z2has a hierarchically hyperbolic structure where the associated hyperbolic spaces are the cosets of the subgroups xay,xbyand the coned-off space S“y CaypZ2qwith respect to xayand xby. The following relations are imposed: •xay K xby •xayĎSand xbyĎS 3. If pX1,S1q,pX2,S2qare HHS, then pX1ˆX2,S1YS2qis a hierarchically hyperbolic space; 4. [14, Theorem 9.1] Let Gbe a group hyperbolic relative to a finite collection Pof peripheral subgroups. If each PPPis a hierarchically hyperbolic group then Gis a hierarchically hyperbolic group. Remark 1.6.4. By [14, Remark 1.3], the projections πUof a hierarchically hyperbolic space pX,Sqcan always be assumed to be uniformly coarsely surjective. Without loss of generality, we will always assume this. Remark 1.6.5. If pX,Sqis a hierarchically hyperbolic space and there exists a metric space Y and a quasi-isometry q:XÑYthen Ycan be endowed with the hierarchical hyperbolic space structure pY,Sq. Indeed, to do so it is enough to keep every element in the index set Sand define projections to every WPSas πW˝q, where qdenotes a quasi-inverse of q. Definition 1.6.6 (Hieromorphism). Let pX,Sqand pX1,S1qbe hierarchically hyperbolic spaces. Ahieromorphism is a triple φ“`φ, φ♦,tφ˚ UuUPS˘, where φ:XÑX1is a map, φ♦:SÑS1is an injective map that preserves nesting, transversality and orthogonality, and, for every UPS, the maps φ˚ U:CUÑCφ♦pUqare quasi-isometric embeddings with uniform constants. 1.7. HIERARCHICALLY HYPERBOLIC GROUPS 17 Moreover, the following two diagrams coarsely commute (again with uniform constants), for all U, V PSsuch that UĎVor U&V: (1.2) Xφ// πU  X1 πφ♦pUq  CUφ˚ U //Cφ♦pUq CUφ˚ U// ρU V  Cφ♦pUq ρφ♦pUq φ♦pVq  CVφ˚ V //Cφ♦pVq 1.7 Hierarchically hyperbolic groups Definition 1.7.1 (Hierarchically hyperbolic group). We say that a group Gis hierarchically hyperbolic if it acts on a hierarchically hyperbolic space pX,Sqsatisfying the following conditions: 1. The action of Gon Xis proper and cobounded; 2. Gacts cofinitely on S(i.e: with finitely many orbits), preserving the relations Ď,Kand &; 3. for each VPSand g, h PG, we have an isometry g:CVÑCgV such that gh :CVÑCghV is the composition of the isometries gand h; 4. for all g1, g2PGwe have associated isometries gi:CVÑCgiVsuch that gπVpxq “ πgV pgxq for every xPXand gρU V“ρgU gV whenever UĹVor U&V. Remark 1.7.2. By definition, if pG, Sqis a hierarchically hyperbolic group and gPG, multiplication by gcoarsely satisfies the two diagrams of Equation (1.2). However, it is always possible to modify the structure to obtain commutativity on the nose, as described in [36, Section 2.1]. This is the reason why the fourth item in Definition 1.7.1 assumes equality. We end this section with a remark/warning: Remark 1.7.3. A hierarchically hyperbolic space may admit several structures. Consider the free group on two generators G“F2pa, bq. Since F2is hyperbolic, pF2,tF2uq is a hierarchically hyperbolic structure. On the other hand, Gsplits as xay ˚ xbyand therefore F2is hyperbolic relative to txay,xbyu. Following the previous theorem we obtain a non-trivial hierarchical hyperbolic structure on F2. To end the chapter, we include various notions and tools exclusive to hierarchically hyperbolic spaces that are are needed to develop the rest of the thesis. 24 CHAPTER 1. PRELIMINARIES (respectively for every WPSw), where pu:GÑGuis the canonical projection on the first direct factor, and πU:GuÑ2CUis the projection given in pGu,Suq. It follows that for every UPSuthe set πUpGwqis uniformly bounded, and analogously for every WPSwthe set πWpGuqis uniformly bounded. Moreover, the inclusions of the subgroups Guand Gwinto Gare full, hierarchically quasiconvex hieromorphisms that induce isometries at the level of hyperbolic spaces. Example 1.9.2 (Free product of hierarchically hyperbolic groups). Let pGu,Suqand pGw,Swqbe hierarchically hyperbolic groups. The free product Gu˚Gwis a hierarchically hyperbolic group. One way of seeing this is to recall that Gu˚Gwis hyperbolic relative to tGu, Gwuand using the following theorem which shows that groups that are hyperbolic relative to a collection of hierarchically hyperbolic subgroups are hierarchically hyperbolic. The proof is already presented in [14, Theorem 9.1], but we describe the structure here to help with the exposition. Theorem 1.9.3. [14, Theorem 9.1] Let Gbe a group relative to a finite collection of peripheral subgroups tH1, . . . , Hku. If each Hican be endowed with a hierarchically hyperbolic group structure, then Gis a hierarchically hyperbolic group. Proof. For each i“1...,n and each left coset of Hiin G, fix a representative gHi. Let gSibe a copy of Siwith its associated hyperbolic spaces and projections in such a way that there is a hieromorphism HiÑgHiequivariant with respect to the conjugation isomorphism HiÑHg i. Let p Gbe the hyperbolic space obtained by coning-off Gwith respect to the peripherals tHiu, and let S“ tp GuYŮgPgŮiSgHi. The relation of nesting, orthogonality or transversality between hyperbolic spaces belonging to the same copy SgHiare the same as in SHi. Further, if U, V belong in two different copies of different cosets, then we impose transversality between them. Finally, for every UPSgHiwe declare that Uis nested into p G. The projections are defined as follows: πp G:GÑp Gis the inclusion, which is coarsely surjective and hence has quasiconvex image. For each UPSgHi, let ggHi:GÑgHibe the closest-point projection onto gHiand let πG U“πHi U˝ggHi, to extend the domain of πUfrom gHito G. Since each πHi Uwas coarsely Lipschitz on CUwith quasiconvex image, and the closest-point projection in Gis uniformly coarsely Lipschitz (Lemma 1.4.6), the projection πG Uis uniformly coarsely Lipschitz and has quasiconvex image. For each U, V PSgHi, the various ρV Uand ρU Vare already defined. If UPSgHiand VPSg1Hj, then ρU V“πVpgg1HjpgHiqq. Finally, for U‰p G, we define ρU p Gto be the cone-point over the unique gHiwith UPSgHi, and ρp G U:p GÑCUis defined as follows: for xPG, let ρp G Upxq “ πG Upxq. If xPp Gis a cone point over g1Hj‰gHi, let ρp G Upxq “ ρSg1Hj U, where Sg1Hjis the Ď–maximal element of Sg1Hj. The cone-point over gHimay be sent anywhere in CU. By [14, Theorem 9.1], the construction above endows pG, Sqwith a hierarchically hyperbolic group 1.9. CONSTRUCTING EXAMPLES OF HIERARCHICAL HYPERBOLICITY 25 structure. Remark 1.9.4. In Theorem 4.2.2 we readapt this theorem to a more general statement. A special type of groups that can be built inductively through direct and free products are known as graph products of groups: Definition 1.9.5. [Graph products] Let Γ be a graph and G“ tGvuvPVpΓqbe a collection of groups. The graph product ΓGwith respect to Gis defined as ΓG“ x˚vPVpΓqGv| rGv, Gws “ 1ô tv, wu P EpΓqy If Gis assumed to be a collection of δ-hyperbolic groups, then by the preceding discussion it is natural to expect that the graph product ΓGis hierarchically hyperbolic. This is indeed the case, and we show a proof of this in Theorem 3.3.7. Even more, we show that graph products of hierarchically hyperbolic groups which have some very natural extra properties (intersection property and clean containers) are hierarchically hyperbolic. We would also like to mention that that in [16], Berlyne and Russel give an independent proof that graph products of hierarchically hyperbolic groups are hierarchically hyperbolic that improves Theorem 3.3.7 by removing the extra assumptions. 1.9.1 Hierarchically hyperbolic structures on groups acting on trees The main contributions of this thesis is the introduction of a wide variety of new examples of hierarchically hyperbolic groups. These are achieved by establishing a combination theorem in this class. If Cis a class of groups, we usually refer to a result as a combination theorem in Cif it provides sufficient conditions ensuring that the fundamental group of a graph of groups in Cis again in C. The Bestvina-Feighn combination theorem [17] for hyperbolic groups is such an example: given a finite graph Gof hyperbolic groups satisfying certain conditions, the resulting fundamental group is again hyperbolic. Their strategy of proof was to consider a metric space (more precisely, atree of metric spaces obtained from the Bass-Serre tree of the graph and the vertex/edge groups of G) and study the action of the fundamental group on such space. This approach turned out to be very successful, and was later applied in several other related contexts. This is the case for the combination theorem of [66] in the class of strongly relatively hyperbolic groups, or for the Hsu-Wise combination theorem in the context of groups acting on cube complexes [51], or Alibegovi´c’s combination theorem for relatively hyperbolic groups [5]. On the other hand, a more dynamical approach is undertaken by Dahmani [29] to obtain another combination theorem for relatively hyperbolic groups. In Chapter 3 we present a combination theorem for hierarchically hyperbolic groups (Theorem 3.0.1). As with the main definition of hierarchical hyperbolicity, understanding the full statement of Theorem 3.0.1 requires the understanding of certain tools and technicalities. Chapter 3 and 2 are dedicated to the development of said tools. We thus postpone the full formulation of the combination theorem to Chapter 3. 26 CHAPTER 1. PRELIMINARIES We now review Example 1.9.2 from a Bass-Serre theory perspective. Example 1.9.6. Let G“Gu˚Gwbe the free product of two hyperbolic groups. We describe here an alternative hierarchically hyperbolic structure on Gfrom that in Example 1.9.2. Let Guand Gwbe endowed with the trivial hierarchically hyperbolic structure. Recall that each vertex vin the Bass-Serre tree Tassociated to Gu˚Gwcorresponds to the set of cosets Pof Gu and Gwin Gu˚Gw. Let Xu, Xwdenote the KpG, 1q-spaces associated to Guand Gwrespectively (i.e the CW-complexes such that π1pXuq “ Guand π1pXwq “ Gw). Recall that the join space X“Xu_Xwis the KpG, 1q-space of Gu˚Gw. π1(Xu) = Guπ1(Xw) = Gw π1(Xu_Xw) = Gu∗Gw The universal cover r Xof Xcan be described as a space which has a combinatorial pattern of an infinite tree. The tree is bipartite with vertices labeled by the symbols Xuand Xw, ( i.e, the Bass-Serre tree of G) as indicated in Figure 1.9.6. Moreover, the number of edges incident on a vertex labelled with Xuare in bijection with π1pXuqand likewise with Xwand π1pXwq. To each vertex labeled with Xu(respectively Xw) we associate the metric space Ă Xu(respectively Ą Xw). This description of r Xcan be thought of as a tree of spaces: Definition 1.9.7. [Tree of spaces] Let Tbe a simplicial tree and let V“VpTq, E “EpTq denote its vertex and edge set respectively. A tree of spaces consists of the quadruple T“ pT, tXvu,tXeu,tφe˘uqvPV,ePE where the maps φe˘:XeÑXe˘are injective functions. If Tis a tree of spaces, we define XpTqthe total space of Tas the metric space where the underlying set is ŮvPVXvand adding edges of length one as follows: if xPXe, we declare φe´pxq to be joined by an edge to φe`pxq. We define the distance on Xas follows: if x, x1are elements on the same vertex space Xv, then we say that dXpx, x1q “ dXvpx, x1q. If x, x1are joined by an edge, we define dXpx, x1q “ 1. Given a sequence x1, . . . , xnof points either joined by an edge or living in the same vertex space, we define its length to be řidXpxi, xi`1q. For general elements 1.9. CONSTRUCTING EXAMPLES OF HIERARCHICAL HYPERBOLICITY 27 f Xu g Xw g Xw f Xu g Xw Figure 1.2: Covering space of Xu_Xw x, x1in X, we define the distance dXpx, x1qas the infimum between all lengths of sequences such that x“x0, . . . , xk“x1. It is not hard to convince oneself that the total space XpTqis quasi-isometric to r X. Indeed, if we collapse each pair of points in XpTqjoined by an edge to a point we obtain r X. This is clearly a quasi-isometry, as all that we have done is collapse uniformly bounded subspaces of XpTqto a point. We now show that Ghas a hierarchical hyperbolic group structure obtained through the action of Gon r X. We begin by describing a hierarchically hyperbolic space structure on r X. (Index set) If Tis the tree of spaces of r Xwe define the index set as S“ tTuYŮvPVtXvu. (Hyperbolic spaces) We declare that CT“Tand that CXv“Xvfor every vertex vin T. (Projections) Note that there is a well-defined map pT:XpTq Ñ Tobtained by collapsing each vertex space to a point. Moreover, it is straightforward to check that this is a coarsely Lipschitz map. We then define the projection pTto be the projection πTfrom XpTqto T. For each xPXpTqand each Xvwe define the closest-point projection pv:XpTq Ñ Xvas follows. Let xPXbe an arbitrary element. If xPXv, then define pvpxq:“x. If xRXv, then we define pvpxqinductively. Let wbe the vertex such that xPXw, suppose that dTpv, wq “ ně1, and that pvp´q is defined on all vertex spaces that are at distance strictly less than nfrom v. Let γbe the geodesic in Tconnecting wto v, let ebe its first edge, with e´“v. It follows that dTpe`, vq “ n´1. Then pvpxq:“pv´φe`˝φe´`pe´pxq˘¯, where φe´is a quasi-inverse of φe´. The various projections ρV Uare defined as follows: First, if U, V correspond to vertex spaces Xu, Xv respectively, then ρU Vis defined as pvpXuqand ρV Uas pupXvq. Note that these are points, as the edge spaces in Tare trivial. If Ucorresponds to Tand Vto a vertex space Xv, we define ρV Uas v. (Relations) For every pair of different vertices v, w we impose that Xv&Xwand that XvĎTfor every vPT. 28 CHAPTER 1. PRELIMINARIES With this structure, all of the axioms of Definition 1.6.1 can be verified. We skip this verification because the structure is simple enough so that all of the axioms are either automatically satisfied or straightforward to prove. Remark 1.9.8. Recall that if we collapse every coset in Pto a point we obtain the coned-off Cayley graph p Gwith respect to tGu, Gwu. Thus, if we define the map p GÑTby sending a coset of Guor Gwto its corresponding vertex in T, then we have a (coarsely)-well defined map. It follows from [67, Lemma 3.1] that this map yields a quasi-isometry between p Gand T. Then, we can simply switch the element Tin the structure defined above by p G. By doing so, we obtain the same structure described in Example 1.9.2 on G. Let us now show that the action of Gon Xsatisfies the axioms of Definition 1.7.1. The first axiom is straightforward to check, as the quotient X{Gis equal to Xv_Xw, which is a compact space. The second one follows from the fact that every vertex space in Xis a copy of either Xvor Xw, which means that there are only finitely many orbits of elements in S. For the third axiom consider gPGand Xva vertex space. Then, g¨v“gv is a vertex in T, and the associated vertex space is Xgv is an isometric copy of Xv. To check the last item, let gbe an element in Gand let xPX. In particular, there is some v1PTsuch that xPXv1. If Xvis a vertex space, then there is a unique path between vand v1in Tthat we call rv, v1s. Let pvdenote the closest-point projection onto a vertex space described above. If eis the last vertex in rv, v1sthen pvpxq “ φe`p˚q, where Xe“ ˚. On the other hand, if we apply gto rv, v1swe obtain the path rgv,gv1sand therefore pgvpgxq “ φpgeq`p˚q “ gφe`p˚q “ gpvpxq. One can argue analogously to obtain that gpvpXv1q “ pgvpXgv1qand, thus, gρU V“ρgU gV for every U, V in ŮvPVXv. If U“Tand V“Xv, then ρV U“vand, therefore, gρV U“ρXgv T“ρgV gU . Remark 1.9.9. The reader may have already noticed that the group G“Gv˚Gwwas known to be hierarchically hyperbolic from the beginning simply because the free product of hyperbolic groups is hyperbolic. This is indeed the case, but we chose to describe this specific structure because Theorem 3.0.1 generalizes this idea. In that sense, the example above is the most basic case possible of Theorem 3.0.1. 1.9.2 A characterization of hierarchical hyperbolicity in hyperbolic-2decomposable groups In the same way as the presence of Z2as a subgroup of Gprevents it from being hyperbolic, the presence of the so-called unbalanced Baumslag–Solitar subgroups prevents Gfrom being hierarchically hyperbolic. The following remark shows this fact: Remark 1.9.10. If Gis a hierarchically hyperbolic group, then Gcannot have a subgroup isomorphic to BSpn, mq “ xa, t |tant´1“amy, with |n| ‰ |m|. Indeed, suppose there is an embedding 1.9. CONSTRUCTING EXAMPLES OF HIERARCHICAL HYPERBOLICITY 29 ι:BSpn, mqãÑG. We have that ιpaqis an infinite order element of G. By [35, Theorem 7.1] and [36, Theorem 3.1], ιpaqis undistorted, which is a contradiction. This prompts the question: is the absence of unbalanced Baumslag–Solitar subgroups in a group Genough to show that Gis hierarchically hyperbolic? This is indeed a very big question without assuming anything on the group. Instead, in this thesis we propose a more reasonable one that assumes that Gsplits over virtually cyclic groups. More precisely, we consider groups that split as graphs of groups with 2-ended edge groups. For the sake of brevity, if Pis a property of a group, we say that a group is P-2-decomposable if it splits as a graph of groups with 2-ended edge groups and vertex groups satisfying property P. Considering groups of this form is not a novelty in geometric group theory. An important example is the class of Z-2-decomposable groups, also known as generalized Baumslag–Solitar groups (GBS groups). Although we will not dive deeply in the theory of GBS groups from a traditional viewpoint, it is worth noting that this class has been extensively studied and shown to be an extremely rich object to analyse from multiple points of view. To name a few, GBS groups have been studied in relation with JSJ decompositions ([39]), quasi-isometries ([70]), automorphisms ([58]) and cohomological dimension ([57]). For a general overview of results on GBS groups we refer to the survey by Robinson ([72]). One way to avoid unbalanced Baumslag–Solitar subgroup in Gis to impose a technical condition on Gcalled balancedeness. A group Gis said to be balanced if for every gPGof infinite order, whenever hgih´1“gjfor some hPGit follows that |i| “ |j|. The notion of balancedness played an important role in the theory of graphs of groups. In [90], the author shows that a free-2-decomposable group is subgroup separable if and only if it is balanced. In [78], the authors extend Wise’s result to (virtually-free)-2-decomposable groups, obtaining quasi-isometrical rigidity for certain balanced groups. In [28] the author studies the relation between possible acylindrical actions of (torsion-free)-2-decomposable groups in connection with balancedness of such groups. A naive conjecture to make is that a hyperbolic-2-decomposable group Gis hierarchically hyperbolic if and only if it is balanced. The last chapter of this thesis is dedicated to prove that, up to some issues with torsion on vertex groups, the conjecture holds (Theorem 4.2.15). In order to formulate the results expressly we introduce the notion of almost Baumslag–Solitar groups: Definition 1.9.11. Let Gbe a group. We say that Gis an almost Baumslag–Solitar group if it can be generated by two infinite order elements a, b PGsuch that the equality bamb´1“bn holds for some n, m. In the particular case where |n|‰|m|we say that Gis an unbalanced almost Baumslag–Solitar group. Note that every almost Baumslag–Solitar group is the quotient of some Baumslag–Solitar group. However, such quotient map may not be an isomorphism. We now recall two results due to Bestvina and Feighn that relate hyperbolicity with almost Baumslag–Solitar subgroups: 30 CHAPTER 1. PRELIMINARIES Theorem 1.9.12 (Amalgams over virtually cyclic groups). Suppose that G“G1˚CG2 is an amalgamated free product where Giis hyperbolic and Cis virtually cyclic. The following conditions are equivalent 1. Cis malnormal in either G1or G2; 2. Gis word hyperbolic; 3. Gdoes not contain BSp1,1q – Z2as a subgroup. Theorem 1.9.13 (HNN extensions over virtually cyclic groups). Let Hbe a hyperbolic group and let Gbe the HNN extension G“ xH, tyover the virtually cyclic subgroups Aand B where tAt´1“B. Then the following are equivalent 1. Gis word hyperbolic; 2. Gcontains no almost Baumslag–Solitar subgroup; 3. for all hPH,|AXBh|ă 8 and either Aor Bis malnormal in H. Using the almost Baumslag–Solitar group terminology, in Section 4.1.4 we present the following generalization of the above theorems to the class of hierarchically hyperbolic groups: Theorem 1.9.14. Let Gbe a hyperbolic-2-decomposable group. Then, Gis hierarchically hyperbolic if and only if it contains no unbalanced almost Baumslag–Solitar subgroups. Detecting almost Baumslag–Solitar subgroups: In general, checking whether a given graph of groups contains an almost Baumslag–Solitar subgroup may be challenging. For this reason, we introduce the notion of balanced edges. An edge eof a graph of groups Gis a balanced edge if for every infinite order element gPGeand hPπ1pG´eq if hgih´1“gjthen |i|“|j|. We then have the following criterion to detect almost Baumslag–Solitar subgroups. Theorem 1.9.15. Let Gbe a graph of groups where none of the vertex groups contain distorted cyclic subgroups. Then π1pGqcontains a non-Euclidean almost Baumslag–Solitar subgroup if and only if Ghas an unbalanced edge. The proof of Theorem 1.9.14 and 1.9.15 can be found in Theorem 4.1.23. 1.9.3 A note on torsion The reader will find that Chapter 4 deals with hyperbolic-2-decomposable groups in two separate settings. Namely, when the group has torsion and when it does not. 1.9. CONSTRUCTING EXAMPLES OF HIERARCHICAL HYPERBOLICITY 31 It is perhaps worth noting that the fundamental group of a graph of virtually torsion-free groups may not be virtually torsion free (even when the edge groups are cyclic), as the following example provided by A. Minasyan shows:2 Example 1.9.16. Let Hbe a group isomorphic to BSp2,3q“xa, b |ba2b´1“a3y. Since His not residually finite, its finite residual RespHq “ ŞKďH,|K:H|ă8 Kis non-trivial. Let aPKbe non-trivial and let Gbe constructed as G“ xH, b | rb, Hs “ 1, b2“1y. Note that Gis virtually torsion free. We then construct the HNN extension Γ “ xG, t |tat´1“aby. We now show that Γ is not virtually torsion-free. First, as HďΓ, we have that RespHq ď RespΓq. Therefore, RespΓqmust be non-trivial. Since RespΓqis a normal subgroup of Γ, we have that tat´1“ab PRespΓq. We thus obtain that a´1pabq “ bPRespΓq. We conclude that b, an element of order two, belongs in every finite index subgroup of Γ and therefore Γ is not virtually torsion-free. For this reason, the main result of Chapter 4 has two formulations depending on the case: Theorem 1.9.17. Let Gbe a hyperbolic-2-decomposable group. The following are equivalent. 1. Gadmits a hierarchically hyperbolic group structure. 2. Gdoes not contain a distorted infinite cyclic subgroup. 3. Gdoes not contain a non-Euclidean almost Baumslag–Solitar group. Moreover, if Gis virtually torsion-free, condition (3) can be replaced by 3’. Gdoes not contain a non-Euclidean Baumslag–Solitar group. A straightforward corollary of this theorem is the following. Corollary 1.9.18. Let G“H1˚CH2where Hiare hyperbolic and Cis virtually cyclic. Then G is a hierarchically hyperbolic group. As final remark, we believe that Item (3’) of Theorem 1.9.17 should be true even without the assumption of Gbeing virtually torsion-free. We refer the reader to the Questions section in Chapter 4 for further discussion. 2https://mathoverflow.net/questions/330632/is-an-hnn-extension-of-a-virtually-torsion-free-group-virtuallytorsion-free. 32 CHAPTER 1. PRELIMINARIES Chapter 2 Structural results The objective of this chapter is to obtain structural results that will be necessary for the development of Chapter 3. Moreover, this chapter introduces a number of tools to analyze hierarchical hyperbolic spaces. The first one is the intersection property (see Definition 2.1.1, and the discussion after the statement of Theorem 3.3.7), which in turn leads to the notion of concreteness. We introduce the latter notion to exclude artificial examples of hierarchically hyperbolic spaces that carry some undesirable features. As we will see in this chapter, the intersection property has a very natural definition, and we conjecture that all hierarchically hyperbolic spaces admit a hierarchically hyperbolic structure with the intersection property (see Question 2.0.1 below). On the other hand, concreteness is more technical, but nevertheless we prove in Proposition 2.1.12 that any hierarchically hyperbolic space with the intersection property can be supposed to be concrete. These properties are of independent interest, and we expect them to be of further use. Clean containers (see Remark 1.6.2), a notion introduced originally by Abbott, Behrstock, and Durham [2], is a technical condition that in the graph of multicurves setting (see Subsection 1.5.2) translates into the following: if VĎSis a subsurface of the surface S, then Vand SzVare disjoint, and any subsurface disjoint from Vis contained into SzV. On the other hand, the intersection property is a condition that we introduce, and in the mapping class group setting means that, given two subsurfaces V, U ĎS, the subsurface VXUis the biggest subsurface of Sthat is contained in both Vand U. The intersection property gives to the index set Sthe structure of a lattice. At this point, it is instructive to notice that both VXUand SzVcould be non-connected subsurfaces of S, and indeed the hierarchically hyperbolic structure with clean containers and the intersection property of the pants decomposition graph GpSqis obtained considering all, possibly non-connected, subsurfaces of S. We are inclined to believe that any hierarchically hyperbolic space admits a hierarchically hyperbolic structure with the intersection property and clean containers: Question 2.0.1. Let pX, dXqbe a hierarchically hyperbolic space. Does there exist a hierarchically hyperbolic structure Ssuch that pX,Sqis a hierarchically hyperbolic space with the intersection property and clean containers? 33 40 CHAPTER 2. STRUCTURAL RESULTS every UPS, and by Definition 1.8.1 the element zbelongs to FSεˆteu. Let s0be the constant associated to the Distance Formula Theorem for the space pX,Sq, and consider sąmaxtε, s0u. There exist K, C ą0 such that (2.4) dpx, zq ď Kÿ UPSttdUpπUpxq, πUpzqquus`C “K¨ ˝ÿ UPSSε ttdUpπUpxq, πUpzqquus`ÿ UPSzSSε ttdUpπUpxq, πUpzqquus˛ ‚`C “Kÿ UPSzSSε ttdUpπUpxq, πUpzqquus`C. Note that dUpπUpxq, πUpzqq ď εfor every UPSzSε. Since sąε, from Equation (2.4) we conclude that dpx, zq ď C. To complete the proof, notice that FSεˆ teucan be endowed with the hierarchical hyperbolic structure SSε. Since X“NCpFSεˆteuq, the space pX,SSεqis hierarchically hyperbolic, being quasi isometric to `FSεˆteu,Sε˘, and it is concrete by construction. The intersection property in pX,SSεqfollows from the intersection property in pX,Sq. Concreteness will play an important role in Lemma 2.3.2 and Theorem 2.3.3, after the proof of Theorem 2.2.1. Lemma 2.1.13. Given a full hieromorphism φ:pX,Sq Ñ pX1,S1q, there exist constants K, C ě0 and s, s1ą0such that ÿ UPSttdUpπUpxq, πUpyqquusďKÿ U1Pφ♦pSqttdU1pπU1pφpxq, πU1pφpyqqquus1`C@x, y PX. Proof. For UPS, we denote φ♦pUqby U1. As the hieromorphism is full, there exists a uniform constant ξsuch that (2.5) dU`πUpxq, πUpyq˘ďξdU1`πU1pφpxqq, πU1pφpyqq˘`ξ, @UPS,@x, y PX. Choose sand s1such that s1:“s´ξ ξą1. Suppose that sďdU`πUpxq, πUpyq˘for a given UPS. Then, using Equation (2.5), we obtain that (2.6) 1 ăs1ďdU1`πU1pφpxqq, πU1pφpyqq˘“ ttdU1`πU1pφpxqq, πU1pφpyqq˘uus1. As sďdU`πUpxq, πUpyq˘we have that ttdU`πUpxq, πUpyq˘uus“dU`πUpxq, πUpyq˘. It then follows 2.1. INTERSECTION PROPERTY AND CONCRETENESS 41 that ttdU`πUpxq, πUpyq˘uus“dU`πUpxq, πUpyq˘ďξdU1`πU1pφpxqq, πU1pφpyqq˘`ξ ďξttdU1`πU1pφpxqq, πU1pφpyqq˘uus1`ξ. (2.7) Therefore, using Equation (2.6) and Equation (2.7), we obtain ttdU`πUpxq, πUpyq˘uusďξttdU1`πU1pφpxqq, πU1pφpyqq˘uus1`ξ ďξttdU1`πU1pφpxqq, πU1pφpyqq˘uus1`ξttdU1`πU1pφpxqq, πU1pφpyqq˘uus1 “2ξttdU1`πU1pφpxqq, πU1pφpyqq˘uus1. (2.8) On the other hand, if sądU`πUpxq, πUpyq˘then (2.9) ttdU`πUpxq, πUpyq˘uus“0ď2ξttdU1`πU1pφpxqq, πU1pφpyqq˘uus1, so the inequality of Equation (2.8) is satisfied also in this case. Concluding, we use Equation (2.8) and Equation (2.9) to obtain that ÿ UPSttdUpπUpxq, πUpyqquusďÿ UPS 2ξttdU1`πU1pφpxqq, πU1pφpyqq˘uus1 “2ξÿ UPSttdU1`πU1pφpxqq, πU1pφpyqq˘uus1 “2ξÿ U1Pφ♦pSqttdU1`πU1pφpxqq, πU1pφpyqq˘uus1, and therefore the lemma is satisfied with K“2ξand C“0. Remark 2.1.14. The argument of Lemma 2.1.13 can be used to show that there exist constants ¯ K, ¯ Cě0 and ¯s, ¯s1ą0 such that ÿ U1Pφ♦pSqttdU1`πU1pφpxqq, πU1pφpyqq˘uu¯sď¯ Kÿ UPSttdUpπUpxq, πUpyqquu¯s1`¯ C@x, y PX. Lemma 2.1.15. Let φ:pX,Sq Ñ pX1,S1qbe a full hieromorphism and Sbe the Ď-maximal element in S. If S1“φ♦pSqand FS1ˆteuis a parallel copy of FS1, then πV1pFS1ˆteuq is coarsely equal to πV1pφpXqq for all V1PS1 S1. Proof. Let zPFS1and consider the tuple ~ b“`πV1pzq˘V1PS1 S1. As zPFS1, the tuple ~ bis κconsistent. The hieromorphism φis full, therefore S1 S1“φ♦pSqand `πV1pzq˘V1PS1 S1“`πV1pzq˘V1Pφ♦pSq. 42 CHAPTER 2. STRUCTURAL RESULTS As the full hieromorphism φinduces uniform quasi isometries ¯ φ˚ V:CV1ÑCVat the level of hyperbolic spaces, we obtain a tuple ~a “ paVqVPS, where aV:“¯ φ˚ V`πV1pzq˘ĎCV. The tuple ~a is κ1-consistent, and therefore there exists xPXthat realizes it, by [14, Theorem 3.1]. Exploiting the fact that the maps φ˚ V˝πVuniformly coarsely coincide with the πV1˝φ(compare Definition 1.6.6) and in particular Equation (1.2)), we conclude that the element φpxqrealizes the tuple ~ b: (2.10) `πV1pzq˘V1Pφ♦pSq—`πV1pφpxqq˘V1Pφ♦pSq. That is, there exists a constant T1depending only on the realization Theorem [14, Theorem 3.1] and the hieromorphism φsuch that dV1pπV1pzq, πV1pφpxqqq ď T1for every V1PS1 S1. Conversely, let φpxq P φpXqand consider the tuple ~c: cV1“$ ’ ’ ’ ’ & ’ ’ ’ ’ % πV1pφpxqq,@V1PS1 S1; πV1peq,@V1PS1K S1; ρS1 V1@V1&S1or V1ĚS1. Since ~c is a κ-consistent tuple, there exists zPXsuch that πVpzq — πVp~c q, and zbelongs to FS1ˆteuby Definition 1.8.1. Therefore there exists T2such that dV1pπV1pzq, πV1pφpxqqq ď T2for every V1PS1 S1. Proposition 2.1.16. If φ:pX,Sq Ñ pX1,S1qis a full hieromorphism between hierarchically hyperbolic spaces, then the spaces Xand FS1are quasi isometric, where S1is the image in S1of the Ď-maximal element of S. Proof. We define a map ψ:FS1ÑXand we prove that it is a quasi isometry. Let zPFS1, and consider the tuple ~ b“`πV1pzq˘V1PS1 S1. As zPFS1, the tuple ~ bis κ–consistent. The hieromorphism φis full, so that S1 S1“φ♦pSqand `πV1pzq˘V1PS1 S1“`πV1pzq˘V1Pφ♦pSq. As the full hieromorphism φinduces uniform quasi isometries ¯ φ˚ V:CV1ÑCVat the level of hyperbolic spaces, we obtain a tuple ~a “ paVqVPS, where aV:“¯ φ˚ V`πV1pzq˘ĎCV. The tuple ~a is κ1-consistent, and therefore there exists xPXthat realizes it by [14, Theorem 3.1]. Exploiting the fact that the maps φ˚ Vuniformly coarsely commute with the projections πV (compare Definition 1.6.6 and in particular Equation (1.2)), we conclude that the element φpxq 2.2. PROOF OF THE MAIN THEOREM 43 realizes the tuple ~ b: (2.11) `πV1pzq˘V1Pφ♦pSq—`πV1pφpxqq˘V1Pφ♦pSq. Define ψpzq:“x. The element xis not uniquely determined by the tuple~ b, but it is up to uniformly bounded error. Let us prove that ψis a quasi isometry. Indeed, let z1, z2PFS1. Using, in this order, the Distance Formula in X1, Remark 2.1.14, and the fact that φis a full hieromorphism combined with the Distance Formula in FS1, we have that (2.12) dXpψpz1q, ψpz2qq ď Kÿ UPSttdUpπUpψpz1qq, πUpψpz2qqquus`C ďK´K1ÿ U1Pφ♦pSqttdU1pπU1pz1q, πU1pz2qquu¯s`C1¯`C ďK`K1pK2dX1pz1, z2q`C2q`C1˘`C. On the other hand, we have that (2.13) dX1pz1, z2q ď K3ÿ U1Pφ♦pSqttdU1`πU1pz1q, πU1pz2q˘uus1`C3 ďK3´K4ÿ UPSttdU`πUpψpz1qq, πUpψpz2qq˘uu¯s1¯`C3 ďK3`K4pK5dXpψpz1q, ψpz2qq`C5q`C4˘`C3. Equation (2.12) and Equation (2.13) prove that ψis a quasi-isometric embedding. Moreover, ψis coarsely surjective. Indeed, given an element xPX, the tuple pπV1pφpxqqV1Pφ♦pSq is consistent, and therefore there exists a point zPFS1coarsely realizing it, that is uniformly close to x. We are now ready to prove the main result of this chapter. 2.2 Proof of the main Theorem Theorem 2.2.1. Let φ:pX,Sq Ñ pX1,S1qbe a full hieromorphism with hierarchically quasiconvex image, and let Sbe the Ď-maximal element of S. The following are equivalent: 1. φis coarsely lipschitz; 2. φis a quasi-isometric embedding; 44 CHAPTER 2. STRUCTURAL RESULTS 3. the maps gφpXq:Fφ♦pSqÑφpXqand gFφ♦pSq:φpXq Ñ Fφ♦pSqare quasi-inverses of each other, and in particular quasi isometries; 4. the subspace φpXq Ď X1, endowed with the subspace metric, admits a hierarchically hyperbolic structure obtained from the one of Xby composition with the map φ; 5. πWpφpXqq is uniformly bounded for every WPS1zφ♦pSq. Proof. The implications 3 ô5ñ1ô2ñ4ñ1 and 2 ñ3 are enough to prove the theorem. 5ñ1 By the Distance Formula applied in pX1,S1q, there exists s0such that for every sąs0 there exists K1, C1ě0 for which (2.14) dX1pφpxq, φpyqq ď K1ÿ VPS1ttdVpπVpφpxqq, πVpφpyqqquus`C1@x, y PX. Also, the Distance Formula applied in pX,Sqimplies that there exists s1such that for every sąs1 there exist K, C ě0 for which (2.15) dXpx, yq ě K´1ÿ UPSttdUpπUpxq, πUpyqquus´C@x, y PX. Now let x, y PX. By hypothesis πWpφpXqq is uniformly bounded for every WPS1zφ♦pSq. Let Mbe this uniform bound, and choose ssuch that sąmaxtM, s0u. Therefore ÿ VPS1ttdVpπVpφpxqq, πVpφpyqqquus“ÿ U1Pφ♦pSqttdU1pπU1pφpxqq, πU1pφpyqqquus and Equation (2.14) implies that dX1pφpxq, φpyqq ď K1ÿ U1Pφ♦pSqttdU1pπU1pφpxqq, πU1pφpyqqquus`C1. Using Remark 2.1.14, we can choose ¯s, ¯s1ąs1and ¯ K, ¯ Cě0 for which ÿ U1Pφ♦pSqttdU1pπU1pφpxqq, πU1pφpyqqquu¯sď¯ Kÿ UPSttdUpπUpxq, πUpyqquu¯s1`¯ C. By taking ˜s“maxts0,¯suwe get ÿ U1Pφ♦pSqttdU1pπU1pφpxqq, πU1pφpyqqquu˜sďÿ U1Pφ♦pSqttdU1pπU1pφpxqq, πU1pφpyqqquu¯s ď¯ Kÿ UPSttdUpπUpxq, πUpyqquu¯s1`¯ C. 2.2. PROOF OF THE MAIN THEOREM 45 As ¯s1ąs1, by the Distance Formula, Equation (2.14) and Equation (2.15) we obtain dX1pφpxq, φpyqq ď K1ÿ U1Pφ♦pSqttdU1pπU1pφpxqq, πU1pφpyqqquu˜s`C1 ďK1¯ Kÿ UPSttdUpπUpxq, πUpyqquu¯s1`K1¯ C`C1 ďK1¯ KpKdXpx, yq`KCq`K1¯ C`C1“RdXpx, yq`R1 for appropriate constants Rand R1. Therefore, φis a coarsely lipschitz map. 1ô2 If φis a quasi-isometric embedding, then it is a coarsely lipschitz map. Suppose now that φis a coarsely lipschitz map. To conclude that it is a quasi-isometric embedding, we need to prove that there exist constants K, C ě0 such that dXpx, yq ď KdX1pφpxq, φpyqq`C for every x, y PX. By the Distance Formula applied in pX,Sq, there exists s0so that for every sěs0there exist K1, C1ě0 so that dXpx, yq ď K1ÿ UPSttdUpπUpxq, πUpyqquus`C1,@x, y PX. Also by the Distance Formula applied to pX1,S1q, there exists s1so that for every sěs1there exist K2, C2ě0 so that dX1pφpxq, φpyqq ě K´1 2ÿ WPS1ttdWpπWpφpxqq, πWpφpyqqquus´C2,@x, y PX. By Lemma 2.1.13, we can choose ¯s, ¯s1ąs1and ¯ K, ¯ Cě0 such that ÿ UPSttdUpπUpxq, πUpyqquu¯sď¯ Kÿ U1Pφ♦pSqttdU1pπU1pφpxqq, πU1pφpyqqquu¯s1`¯ C ď¯ Kÿ WPS1ttdWpπWpφpxqq, πWpφpyqqquu¯s1`¯ C, @x, y PX. Let s“maxts0,¯su. Since sěs0and sě¯s, for any x, y PXwe obtain that dXpx, yq ď K1ÿ UPSttdUpπUpxq, πUpyqquus`C1ďK1ÿ UPSttdUpπUpxq, πUpyqquu¯s`C1 ďK1˜¯ Kÿ WPS1ttdWpπWpφpxqq, πWpφpyqqquu¯s1`¯ C¸`C1 ďK1¯ K`K2dX1pφpxq, φpyqq` ¯ KC2˘`K1¯ C`C1“SdX1pφpxq, φpyqq`S1 for appropriate constants Sand S1. Therefore, φis a quasi-isometric embedding. 46 CHAPTER 2. STRUCTURAL RESULTS 2ñ4 If the map φis a quasi-isometric embedding then p4qis automatically satisfied, because hierarchical hyperbolicity is preserved under quasi isometries (compare with the remark before [12, Theorem G]). 4ñ1 As the hieromorphism is full, every induced map φ˚ U:CUÑCpφ♦pUqq is a pξ, ξq-quasi isometry, where ξis independent of UPS, that is ξ´1dUpπUpxq, πUpyqq´ξďdφ♦pUqpφ˚ UpπUpxqq, φ˚ UpπUpyqqq ď ξdUpπUpxq, πUpyqq`ξ for all UPSand for all x, y PX. By the Distance Formula applied in pX,Sq, there exists s0such that for every sěs0there exist K1, C1ě0 satisfying (2.16) dXpx, yq ě K´1 1ÿ UPSttdUpπUpxq, πUpyqquus´C1,@x, y PX. We apply now the Distance Formula to the hierarchically hyperbolic space pφpXq, φ♦pSqq. Therefore, there exists s1such that for every sěs1there exist K2, C2ě0 satisfying (2.17) dX1pφpxq, φpyqq ď K2ÿ U1Pφ♦pSqttdU1pπU1pφpxqq, πU1pφpyqqquus`C2,@x, y PX. By Remark 2.1.14, we can choose ¯s, ¯s1ąs0and ¯ K, ¯ Cě0 for which (2.18) ÿ U1Pφ♦pSqttdU1pπU1pφpxqq, πU1pφpyqqquu¯sď¯ Kÿ UPSttdUpπUpxq, πUpyqquu¯s1`¯ C, @x, y PX. For s“maxts1,¯su, combining Equation (2.16), Equation (2.17), and Equation (2.18), we obtain that dX1pφpxq, φpyqq ď K2ÿ U1Pφ♦pSqttdU1pπU1pφpxqq, πU1pφpyqqquus`C2 ďK2˜¯ Kÿ UPSttdUpπUpxq, πUpyqquu¯s1`¯ C¸`C2 ďK2¯ KpK1dXpx, yq`K1C1q`K2¯ C`C2“TdXpx, yq`T1 for appropriate constants Tand T1. Therefore, φis a coarsely lipschitz map. 3ñ5By hypothesis, gFS1:φpXq Ñ FS1and gφpXq:FS1ÑφpXqare quasi inverses of each other, and by construction of gate maps they are also coarsely lipschitz. Therefore FS1and φpXqare quasi-isometric, where the quasi-isometry is given by gFS1, and in particular there exists Cą0 such that 2.2. PROOF OF THE MAIN THEOREM 47 φpXq Ď NCpgφpXqpFS1qq. Let WPS1zφ♦pSq. By the previous inclusion, there exists C1ą0, depending on Cand on πW, such that (2.19) πWpφpXqq Ď NC1`πWpgφpXqpFS1qq˘. Since the hieromorphism φis full, φ♦pSq “ S1 S1. Moreover, by construction of gate maps, the set πWpgφpXqpFS1qq is uniformly coarsely equal to pπWpφpXqqpπWpFS1qq, where pπWpφpXqq is the closest-point projection in CWto the quasiconvex subspace πWpφpXqq. Since WPS1zS1 S1, we have that diampπWpFS1qq ď αby [14, Construction 5.10] and, as a consequence, that there exists α1such that diampπWpgφpXqpFS1qqq ď α1. The first condition of the theorem follows from this, and Equation (2.19). 5ñ3We claim that there exists Mą0 such that dX1pgFS1˝gφpXqpzq, zq ď M, dX1pgφpXq˝gFS1pyq, yq ď M, @zPFS1,@yPφpXq. By applying the Distance Formula to the space pX1,S1q, there exists s0such that for every sěs0 there exist K1, C1ą0 such that dX1pgFS1˝gφpXqpzq, zq ď K1ÿ U1PS1ttdU1pπU1pgFS1˝gφpXqpzqq, πU1pzqquus`C1,@zPFS1. By Lemma 2.1.7, diampπWpFS1qq ď εfor every WPS1zS1S1for an appropriate εą0. For sěmaxts0, εuand the previous equation, it follows that (2.20) dX1pgFS1˝gφpXqpzq, zq ď K1ÿ U1PS1 S1 ttdU1pπU1pgFS1˝gφpXqpzqq, πU1pzqquus`C1,@zPFS1. For zPFS1, using the fact that gFS1pzq “ z, we obtain (2.21) dU1`πU1pgFS1˝gφpXqpzqq, πU1pzq˘“dU1pπU1pgFS1˝gφpXqpzqq, πU1pgFS1pzqqq ď ďdU1pppπU1˝gφpXqpzqq, ppπU1pzqqq`2k ďk1dU1pπU1pgφpXqpzq, πU1pzqq`c1`2k, where p:CU1ÑπU1pFS1qis the closest-point projection to the quasiconvex subspace πU1pFS1q Ď CU1, and k1, c1denote the multiplicative and additive constants associated to the coarsely lipschitz map p, and kdenotes the Hausdorff distance between the (uniformly) coarsely equal sets 48 CHAPTER 2. STRUCTURAL RESULTS πWpgFS1pxqq and ppπWpxqq, for every xPX1. By Lemma 2.1.15 there exists a constant Tą0 such that for every zPFS1there exists φpxq P φpXq for which dU1pπU1pφpxqq, πU1pzqq ď Tfor every U1PS1 S1. Since πU1pgφpXqpzqq coarsely equals pπU1pφpXqqpπU1pzqq, we obtain that dU1pπU1pgφpXqpzqq, πU1pzqq ď T1@U1PS1 S1. By choosing an adequate sin Equation (2.20), we conclude that dX1pgFS1˝gφpXqpzq, zq ď C1. In order to show that dX1pgφpXq˝gFS1pyq, yqis uniformly bounded for every yPφpXqlet µą0 denote the constant such that diampπWpφpXqqq ă µfor every WPS1zφ♦pSq “ S1zS1 S1. By the Distance Formula there exists s0ą0 such that for all sěs0there exists K2, C2such that (2.22) dX1pgφpXq˝gFS1pyq, yq ď K2ÿ U1PS1ttdU1pπU1pgφpXq˝gFS1pyqq, πU1pyqquus`C2,@yPφpXq. Since πU1˝gφpXq—pπU1pφpXqq ˝πU1, it follows that πU1pgφpXq˝gFS1q — pπU1pφpXqqpπU1˝gFS1q. Moreover, if U1ĎS1, it follows that πU1˝gFS1—πU1, because πU1pFS1q — πU1pX1qfor every U1ĎS1. Therefore, we conclude that πU1pgφpXq˝gFS1q — pπU1pφpXqq˝πU1. For any yPφpXqwe have that pπU1pφpXqq ˝πU1pyq “ πU1pyqand, therefore, πU1pgφpXq˝gFS1pyqq — πU1pyqfor every U1PS1 S1, that is for all U1PS1and for all yPφpXq, we have that dU1pπU1pgφpXq˝gFS1pyqq, πU1pyqq ď ¯µfor some constant ¯µ. For sěmaxts0, µ, ¯µu, Equation (2.22) yields that dpgφpXq˝gFS1pyq, yq ď C2, that is the distance is uniformly bounded. 2ñ3 We claim that pφpXq,S1S1qis a hierarchically hyperbolic space. Since pX,Sqis a hierarchically hyperbolic space and φpXqis quasi isometric to X, we can endow φpXqwith the hierarchically hyperbolic structure given by the index set S. For VPS, the projections πV:φpXq Ñ CVin this latter hierarchically hyperbolic space are defined to be πV˝φ´1, where φ´1is a fixed quasi inverse of φ:XÑφpXq, and πVare the projections in the space pX,Sq. Moreover, we can define the hierarchically hyperbolic space pφpXq, φ♦pSqq. For V1Pφ♦pSq, that is for V1“φ♦pVqwith VPS, the projections πV1:φpXq Ñ CV1are defined to be φ˚ V˝πV˝φ´1, where φ´1and πVare as before, and φ˚ V:CVÑCV1are the (uniform) quasi isometries provided by the hierarchically hyperbolic space pX,Sq. By Definition 1.6.6 we have that φ˚ V˝πV—πV1˝φ, where πV1is the projection in the space pX1,S1q. Therefore πV1—πV1˝φ˝φ´1, which uniformly coarsely coincides with πV1, being φand 2.3. MAIN STRUCTURAL RESULTS 49 φ´1quasi inverses of each other. Thus pφpXq, φ♦pSqq is a hierarchically hyperbolic space, where we can take the projections to be πV1for all V1Pφ♦pSq, instead of πV1. From this point, the argument to prove that there exists Mą0 such that dX1pgFS1˝gφpXqpzq, zq ď M, dX1pgφpXq˝gFS1pyq, yq ď M@zPFS1, y PφpXq is exactly the same as the one used in the previous implication 5 ñ3, and it is omitted. 2.3 Main structural results Theorem 2.2.1 has several consequences. We start with the following: Remark 2.3.1. The combination theorem of Behrstock, Hagen, and Sisto [14, Theorem 8.6] holds without the first part of their fourth hypothesis, that is if eis an edge of Tand Seis the Ď-maximal element of Se, then for all VPSe˘, the elements Vand φ♦ e˘pSeqare not orthogonal in Se˘. Indeed, this hypothesis is used (compare [14, Definition 8.23]) to define the uniformly bounded sets ρrWs rVswhen rWsand rVsare transverse equivalence classes whose supports do not intersect. By Theorem 2.2.1, instead of defining ρrWs rVs“cV˝ρφ♦ e`pSq Vv as done in [14, Definition 8.23], we can impose that ρrWs rVs“cV`πVe`pφe`pXeqq˘, where eis the last edge in the geodesic connecting TrWsto TrVs, with e`PTrVs, and cVis the comparison map from CVe`to the favorite representative of rVs. We will exploit this fact in the proof of Theorem 3.0.1 (compare Subsection 3.2.3 and Equation (3.14)). The proof of [14, Theorem 8.6], after this modification, is not altered. Lemma 2.3.2. Let φ:pX,Sq Ñ pX1,S1qbe a full, coarsely lipschitz hieromorphism between hierarchically hyperbolic spaces such that φpXqis hierarchically quasiconvex in X1, and let Sbe the Ď-maximal element of S. There exist εand ε0such that for all ε1ěε0, if pX,Sqis ε1-concrete, with the intersection property and clean containers, then for any element WPS1we have that WKsuppε`φpXq˘if and only if WKφ♦pSq. 56 CHAPTER 3. A COMBINATION THEOREM Assume first that uand vare vertices connected by a single edge esuch that u“e´and v“e`. Then, the comparison map is defined as c:“φ˚ e`˝φ˚ e´:CVuÑCVv. Where the maps φ˚ e`:CVeÑCVe`and φ˚ e´:CVeÑCVe´are the quasi-isometries induced by the hieromorphisms φe`:XeÑXe`and φe´:XeÑXe´respectively and φ˚ e´denotes a quasi inverse of φ˚ e´. For the general case, let γbe the geodesic in Tconnecting uto v, let uibe the i-th vertex of this geodesic (so that u“u0and v“unfor some natural number ną0), and let eibe the edge connecting ui´1to ui. For all i“1, . . . , n consider the hieromorphisms φe´ i:XeiÑXui´1 and φe` i:XeiÑXui, and the induced quasi-isometries φ˚ e´ i :CVeiÑCVui´1and φ˚ e` i :CVeiÑCVui from the hyperbolic space associated to the representative of rVsin Seito the hyperbolic spaces associated to Vui´1and Vuirespectively. Finally, let φ˚ e´ i :CVui´1ÑCVeibe a quasi-inverse of the map φ˚ e´ i , for all i. Then, the comparison map cis defined to be the composition of the previous quasi isometries: (3.2) c:“φ˚ e´ n˝φ˚ e´ n¨¨¨˝φ˚ e´ 1˝φ˚ e´ 1 :CVu0ÑCVun. Remark 3.1.5. It is a fact [14, Lemma 8.18] that if the cardinality of supports is uniformly bounded, then comparison maps are pξ, ξq-quasi-isometries, for some uniform (not depending on the two vertices uand v) constant ξě1. Remark 3.1.6. If the edge hieromorphisms tφe˘uePEof the tree of hierarchically hyperbolic spaces Tinduce isometries at the level of hyperbolic spaces, then we can choose inverse isometries for the maps φ˚ e˘. Therefore, from Equation (3.2) it follows that comparison maps in this particular case are isometries. We record now the following lemma, which is implicitly used in [14]. Its proof follows by applying repeatedly the (coarsely commutative) second diagram of Equation (1.2). Lemma 3.1.7. Let Tbe a tree of hierarchically hyperbolic spaces, and let rUs,rVsbe two equivalence classes such that either rUs&rVsor rUsĎrVs. If comparison maps are uniform quasi isometries, then for all vertices u, v PTrUsXTrVsthe set cpρUu Vuqis coarsely equal to ρUv Vv, where c:CVuÑCVvis the comparison map. 3.1. TREES OF HIERARCHICALLY HYPERBOLIC SPACES 57 3.1.1 Trees with decorations Recall that a tree of hierarchically hyperbolic spaces (as defined in Definition 3.1.1) is a tuple (3.3) T“´T, tpXv,SvquvPV,tpXe,SequePE˘,tφe˘:pXe,SeqÑpXe˘,Se˘qu¯, where T“ pV, Eqis a tree, tpXv,SvquvPVuand tpXe,SequePEuare families of uniformly hierarchically hyperbolic spaces, and φe`:pXe,Seq Ñ pXe`,Se`qand φe´:pXe,Seq Ñ pXe´,Se´qare hieromorphisms with constants all bounded uniformly. Recall that, on ŮvPVSvone defines the following equivalence class: given an edge e“ tv, wu P E and UPSe, impose φ♦ vpUqto be equivalent to φ♦ wpUq, and take the transitive closure of this to obtain the desired equivalence relation. Given UPŮvPVSv, its equivalence class is denoted by rUs. In general, in a tree of hierarchically hyperbolic spaces Tit might happen that two distinct equivalence classes rUs ‰ rVsare supported on exactly the same vertices of the tree T, that is TrUs“TrVs. This is not desirable, and in this subsection we describe a slight modification of the tree T(and therefore of the metric space XpTqassociated to it) that ensures that rUs “ rVsif and only if TrUs“TrVs. We achieve this by attaching to each vertex vof Ta tree of uniformly bounded diameter, and refer to these attached trees as decorations. We denote the tree that is obtained with this process by r T. As a consequence, the new support trees r TrUswill become larger than the original ones (i.e. TrUsĎr TrUsfor each equivalence class rUs). All the hypotheses of Theorem 3.0.1 are preserved by adding these decorated trees (furthermore, the metric spaces associated to the two trees of hierarchically hyperbolic spaces are quasi-isometric), and therefore for the proof of the theorem we will assume without loss of generality that equivalence classes are discriminated by their supports. We now describe how to decorate the tree Tof hierarchically hyperbolic spaces of Equation (3.3), to ensure that rUs“rVsif and only if TrUs“TrVs. For any vertex vPT, let Svbe the Ď-maximal element in Sv, let Ube any Ď-maximal element of SvztSvuand let FUˆtfube a parallel copy of the FUinside of Xv. For any such choice, we add a new vertex ˜vand a new edge ˜econnecting vand ˜v. The metric spaces X˜vand X˜eare defined to be FUˆtfu, with the induced metric. It follows from [14, Proposition 5.11] that `Xrv,SU˘and `Xre,SU˘are hierarchically hyperbolic spaces, of complexity strictly lower than `Xv,Sv˘. We refer to these index sets as SU,f r vand SU,f r e respectively, where the exponent is added to keep track of the choices of the Ď-maximal element UPSvztSvu, and of the parallel copy FUˆtfu. The hieromorphisms φr e`and φr e´are defined as follows. At the level of metric spaces, φr e`:XreÑ Xrvis the identity map and φre´:XreÑXvis the subspace inclusion. The map φ♦ re`:SU,f r eÑSU,f r vis the identity of the set SU, and φ♦ re´:SU,f reÑSvis the inclusion. At the level of hyperbolic spaces, the maps φ˚ re´,W , φ˚ re`,W :CWÑCWare the identity for each WPSU,f re. It is straightforward to check that the commutative diagrams of Definition 1.6.6 are satisfied. Furthermore, since φ♦ re`, φ♦ re´and φ˚ re`,W , φ˚ re´,W are identity maps or inclusions, it follows that φre`and φre´are full 58 CHAPTER 3. A COMBINATION THEOREM hieromorphism. Moreover, they are quasiconvex. We repeat this process for any newly produced vertex, until the complexity of the resulting hierarchically hyperbolic spaces is one. In particular, given a new vertex r vwith associated hierarchically hyperbolic space `FUˆ tfu,SU˘not of complexity one, consider a Ď-maximal element VPSUztUu. Consider moreover a parallel copy FVˆ tf1uof FVin FUˆ tfu, and repeat the process to construct a new vertex with associated hierarchically hyperbolic space `FVˆtf1u,SV˘. We stress that FVis defined in the hierarchically hyperbolic space `FUˆtfu,SU˘, and not in the space `Xv,Sv˘for which UPSv. We denote by r Tthe tree of hierarchically hyperbolic spaces obtained from Tfollowing this process. Notice that XpTqcan be naturally seen as a subspace of Xpr Tq, that is XpTq Ď Xpr Tq. Moreover, as the complexity of the hierarchically hyperbolic spaces of Tis uniformly bounded and each step of the described process reduces the complexity by one, there exists a uniform constant Csuch that NC`XpTq˘“Xpr Tq. In particular, the inclusion map ι:XpTqãÑXpr Tqis a quasi isometry, and therefore the two spaces XpTqand Xpr Tqare quasi isometric. In Xpr Tq, we denote by „‹the equivalence relation described in Subsection 3.1, by rUs‹the equivalence class of UPŮrvPr VSrvwith respect to „‹, and by r TrUs‹the support of rUs‹. Notice that r TrUs‹XT“TrUsfor all UPŮvPVSv, and that for all r VPŮrvPr VzVSrvthere exists VPŮvPVSv such that r V„‹V. Remark 3.1.8. In the context of hierarchically hyperbolic groups, decorating a tree Tamounts to the following. Let vbe a vertex in Twith associated group G, and consider the Bass-Serre tree of G˚HH, where His a hierarchically quasiconvex subgroup of Gof maximal, strictly smaller complexity, and the two edge-embeddings are given by the identity map idH:HÑHand by the inclusion ι:HÑG. This Bass-Serre tree has one vertex v0with associated group G, and rG:Hs vertices viwhose associated groups are the G-cosets of the subgroup H, and edges eiconnecting v0to vi. In the tree T, we replace the vertex vby v0, and we add new vertices viand edges eiconnecting v0to vi. To these new vertices v0and vi, we associate the groups given by the Bass-Serre tree of the splitting G˚HH. For any new vertex viadded in such way, we repeat the process unless the vertex group Hhas complexity one. Lemma 3.1.9. In the tree of hierarchically hyperbolic spaces r Twe have that rUs‹“ rVs‹if and only if r TrUs‹“r TrVs‹. Proof. One implication is trivial. Assume now that r TrVs‹“r TrWs‹. If the complexity of the two equivalence classes rVs‹and rUs‹is different, then the decorations added to the tree Tare trees of different diameter, and therefore we cannot have that r TrVs‹“r TrWs‹. Thus, the equivalence classes have the same complexity, so neither cannot be properly nested into the other. By construction, in the tree r Tthere are vertices ruand rvsuch that Uand Vare Ď-maximal 3.1. TREES OF HIERARCHICALLY HYPERBOLIC SPACES 59 elements of Sruand Srv, respectively. As r TrUs‹“r TrVs‹, the equivalence class rUs‹must have a representative in Srv, and rVs‹must have a representative in Sru. As neither equivalence class can be properly nested into the other, it must then be that rUs‹“ rVs‹. If the tree Tsatisfies the hypotheses of Theorem 3.0.1, then also r Tdoes. We prove this in the following lemmas. Lemma 3.1.10. In the tree of hierarchically hyperbolic spaces r Tthe edge hieromorphisms are full, coarsely lipschitz, and hierarchically quasiconvex. Proof. Let ebe an edge in r T. Two cases can occur: either eis an edge already in the tree T, or it was added with the decoration of T. If ewas already an edge in T, then the edge hieromorphisms are full, coarsely lipschitz, and hierarchically quasiconvex by the hypotheses of Theorem 3.0.1. On the other hand, if eis a new edge then the two maps φe´and φe`are full, hierarchically quasiconvex isometric embeddings (one is actually an isometry), by construction. Lemma 3.1.11. The hierarchically hyperbolic spaces of r Thave the intersection property and clean containers. Proof. Let rvbe a vertex of r T. If rvPTthen Srvhas the intersection property and clean containers, by the hypotheses of Theorem 3.0.1. If rvPr TzT, then Sr v“SU,f r vcoincides with SU, for some UPŮvPVSv. Therefore, Srvhas in intersection property. Let vPTbe the vertex such that UPSv. Suppose that Srv“SU,f rv“SUdoes not have clean containers. Therefore, there exists WP SUztUusuch that the set tZPSU|ZKWuis not empty, and WMcontU KW. By Lemma 2.1.5 we know that contU KW“U^contKW, where contKWis the orthogonal container of Win Sv. Moreover WKcontKWby clean containers in Sv, and therefore we reach a contradiction, as contU KWĎcontKU. Thus, SU,f rvhas clean containers. The argument for edge spaces is similar. Lemma 3.1.12. Comparison maps in r Tare uniformly quasi-isometries. Proof. Let v, w be two vertices in r Tand let rVs‹be an equivalence class supported on both vertices, with representatives Vvand Vwrespectively. Consider the comparison map c:CVvÑCVw, as defined in Equation (3.2). If both vertices already belong to TĎr T, then the map cis a uniform quasi-isometry by the hypotheses of Theorem 3.0.1. If one vertex, say w, belongs in r TzT, and vPT, consider the geodesic σin r Tconnecting vto w. Let v“v0,...vn“wbe the vertices of σ, such that viis joined by an edge to vi`ifor all i“0, . . . , n ´1. Then, there exists a maximal index i‹such that vi‹PTand vi‹`1Pr TzT; let 60 CHAPTER 3. A COMBINATION THEOREM V‹be the representative of rVsin Svi‹. From Equation (3.2) we see that cis the composition of c1:CVvÑCVvi‹with c2:CVvi‹ÑCVw. As noticed in the previous case, the map c1is a uniform quasi-isometry. Moreover, by construction, the map c2is an isometry, and therefore cis a uniform quasi-isometry, being the composition of these two maps. The last case to consider is when both vertices belong to r TzT. Depending on whether the geodesic σdoes not intersect T, or does intersect it, the map cwill be an isometry, or a composition of three maps, two of which isometies and the remaining a uniform quasi isometry. Therefore, all comparison maps are uniform quasi isometries. In view of this, for the whole proof of Theorem 3.0.1 we assume without loss of generality that equivalence classes are differentiated by their supports already in the tree of hierarchically hyperbolic space T, that is rUs“rVsif and only if TrUs“TrVs. On the other hand, for the proof of Corollary 3.3.1, that is the application of Theorem 3.0.1 to hierarchically hyperbolic groups, we will not decorate the tree T. This is because, even if a hierarchically hyperbolic group `G, S˘acts on the index set S, the set of product regions FUˆ tfu | UPS, f PEU(might not be G-invariant. Therefore, it might happen that the hierarchically hyperbolic space pXpr Tq,r Sq, where r Sdenotes the index set associated to the decorated tree r T, does not admit a non-trivial action of Gonto r S. We refer to Section 3.3 for the complete treatment of this delicate point. We now define the hierarchically hyperbolic structure on this tree of hierarchically hyperbolic spaces. 3.2 Endowing a tree of HHS with an HHS structure As we have seen, whenever we are presented with a tree of metric spaces T, it is possible to associate a metric space XpTqto it called the total space of T. Theorem 3.0.1 gives sufficient conditions under which the total space of a tree of hierarchically hyperbolic spaces has a hierarchically hyperbolic space structure. This section is dedicated to the proof of Theorem 3.0.1 and is divided into three subsections. In the first section we show how the index set is built; the second one describes what the hyperbolic spaces associated to each element in the index set are. Finally, in the last subsection we prove Theorem 3.0.1 with the newly-developed elements. 3.2.1 Construction of index set Remark 3.2.1 (Concreteness of the edge spaces). In the proof of Theorem 3.0.1 we will need to exploit concreteness of the edge spaces, which is not an hypothesis of the theorem. We now explain why we can suppose, without loss of generality, that all the hierarchically hyperbolic edge-spaces of Tare ε-concrete. Let εě3 maxtα, ξuas in Lemma 2.1.7. If the edge spaces are not all ε-concrete, then we apply 3.2. ENDOWING A TREE OF HHS WITH AN HHS STRUCTURE 61 Proposition 2.1.12 to each edge space Seof Tto obtain a sub-index set Se,ε ĎSesuch that pXe,Se,εqis ε-concrete. Notice that if Seis already ε-concrete, then Se,ε “Se. Similarly to what is defined in Subsection 3.1, define „εto be the transitive closure of „d,ε: for any edge eand any UPSe,ε, we have that φe`pUq „d,ε φe´pUq. Doing so (and not defining equivalence classes with respect to the equivalence class „of Subsection 3.1) will be crucial to be able to apply Lemma 2.3.4 during the proof of Theorem 3.0.1. Moreover, this does not affect the hypotheses of the theorem, that continue to be satisfied. Indeed, edge spaces continue to be uniformly hierarchically quasiconvex in vertex spaces, with edge hieromorphisms being full and uniformly coarsely lipschitz. Comparison maps are not affected by this change (but there might be fewer of them, as we are considering possibly smaller edge-space index sets). Finally, the intersection property is preserved by Proposition 2.1.12, and clean containers are preserved by Lemma 2.1.5. In view of Remark 3.2.1, from now on we assume without loss of generality that all edge spaces are ε-concrete for some appropriate ε, that is that the equivalence relations „εand „are the same. Let p Tbe the result of coning off the underlying tree associated to the tree of spaces Twith respect to every support tree TrVs. We define the index set Sassociated to the tree of hierarchically hyperbolic spaces Tas (3.4) S“S1\S2\tp Tu. The set S1is (3.5) S1:“´ğ vPV Sv¯{ „, as defined in Subsection 3.1. Elements of S2correspond to supports of elements in S1: (3.6) S2:“ tTrVs| rVs P S1u. We stress that all these elements are subtrees of the tree T, the tree attached to the tree of hierarchically hyperbolic spaces T. By the following lemma, the set S2is closed under intersections. Lemma 3.2.2. Suppose that TrUsXTrVsis not empty. Then there exists rAs P S1for which TrAs“TrUsXTrVsand rUs,rVsĎrAs. Proof. Let Vvand Uvbe the representatives of rVsand rUsin the index set Sv, for all vP TrUsXTrVs. 62 CHAPTER 3. A COMBINATION THEOREM For all vPTrUsXTrVs, consider the set Λv“ tWPSv|Vv, UvĎWu, which is non-empty since it contains the maximal element of Sv. Since Vv_Wvis, by definition, the Ď-minimal element of Svcontaining both Vvand Wv, it is the unique Ď-minimal element of Λv, which we denote also by Av. If TrUsXTrVsconsists of just one vertex v, then rAs“rVv_Uvsis the desired equivalence class: as rVvsand rUvsare nested into rAs, it follows that TrAsĎTrVsXTrUs. Therefore TrAs“TrVsXTrUs. If TrVsXTrUshas more than one vertex, analogously to what constructed in the index sets of the vertices, there is a unique Ď-minimal element in the edge-index set Sethat we denote by Ae, where eis any edge that contains representatives of both rUsand rVs. Assume now that v, w PTrUsXTrVsand that there is an edge ethat connects these two vertices. Then φ♦ vpAeq “ Avand φ♦ wpAeq “ Aw. Therefore φ♦ vpAeq “ φ♦ vpVe_Ueq “ φ♦ vpVeq_φ♦ vpUeq “ Vv_Uv“Av by Lemma 2.1.3. Thus Av„Awfor all v, w PTrUsXTrVs, and we denote by rAsthe equivalence class of (any of the) rAvs. By construction, rAshas a representative where both rVsand rUshave, and hence TrUsXTrVsĎTrAs. On the other hand we have that rVsand rUsare nested in rUv_Vvs“rAs, and therefore TrAsĎ TrUsXTrVsby Lemma 3.1.2. Thus, the lemma is proved. Corollary 3.2.3. Let rVs,rWsbe equivalence classes. Then, rVsĎrWsif and only if TrWsĎTrVs. Proof. If rVsĎrWsthen TrWsĎTrVs, by Lemma 3.1.2. On the other hand, if TrWsĎTrVswe can see that TrWs“TrWsXTrVs. By Lemma 3.2.2 there exists rAs P S1for which TrAs“TrWsXTrVs and rVs,rWsĎrAs. It follows that TrWs“TrAs, and therefore that rWs“rAs, because we are assuming that the tree Tis decorated (compare Lemma 3.1.9). Thus rVsĎrWs. To define nesting, orthogonality, and transversality, we proceed as follow. The element p Tis the Ď-maximal element. Relations in S1are as in [14]: two „-equivalence classes rVsand rWsare nested (respectively orthogonal), rVsĎrWs(respectively rVs K rWs), if there exist a vertex vPTand representatives Vv, WvPSvsuch that rVs“rVvs,rWs“rWvsand VvĎWv(respectively VvKWv) in Sv. If rVsand rWsare not orthogonal and neither is nested into the other, then they are transverse: rVs&rWs. 3.2. ENDOWING A TREE OF HHS WITH AN HHS STRUCTURE 63 Relations in S2are as follows. For two elements TrVs, TrUsPS2, if TrVsis contained as a set in TrUsthen TrVsĎTrUs, and vice versa. Otherwise they are transverse, TrVs&TrUs. Relations between an equivalence class rWsand an element TrVsPS2are as follows: pc1qif rWsĎrVswe declare rWs K TrVs; pc2qif rWs K rVswe declare rWsĎTrVs; pc3qotherwise, we declare rWs&TrVs; Notice that rWs K TrVsif and only if TrVsĎTrWs, by Corollary 3.2.3. 3.2.2 Hyperbolic spaces associated to the index set and projections Let Cˆ T“ˆ T, which is produced from the tree Tby coning-off each subtree TrWsPS2. Remark 3.2.4. As soon as there exists a vertex space pXv,Svqand two orthogonal elements UKV in Sv, then the decoration trick of Subsection 3.1.1 implies that all supports trees TrWsPS2are properly contained into the tree T. Indeed, if TrWs“Tfor some equivalence class, it must then be that TrUsand TrVsare properly nested into TrWs, and thus rWsĎrUsand rWsĎrVsby Lemma 3.1.9. This contradicts the fact that rUs K rVs, and in particular that there is no equivalence class nested into both. To each equivalence class rVswe associate a favorite vertex vPTrVsand the favorite representative VvPSv, so that rVs“rVvs. Then, define CrVsto be CVv. By assumption, there exists a uniform constant ξě1 such that for all vertices wsuch that there exists WPSwwith W„Vv, the comparison map c:VvÑWis a pξ, ξq-quasi-isometry. For TrWsPS2, let CTrWs:“p TrWsbe the hyperbolic space obtained from the tree TrWsby coning-off each subtree TrVsPS2properly contained in TrWs, that is TrVsĹTrWs. Define πp T:XpTq Ñ p Tas follows: for xPXv, define πp Tpxq “ v. Notice that πp Tis the composition of the projection XÑTof Xon its Bass-Serre tree with the inclusion of the tree Tinto p T. For all TrWsPS2the projection πTrWsis defined analogously: for xPXv, consider the closest-point projection of the vertex vonto the subtree TrWsin the tree T. The image of this point under the inclusion map TãÑp Tis πTrWspxq P CTrWs“p TrWs. These projection maps πTrWsand the projection map πp Tare uniformly coarsely surjective, being surjective on the set of non-cone points. Given rVs P Swith favorite representative V˜vPS˜v, we define πrVs:XÑCrVsas follows. If πp Tpxq “ vis a vertex in the support of rVs, then there exists a representative VvPSvof the class rVs, and πrVspxqis defined to be (3.7) πrVspxq:“c˝πVvpxq Ď CV˜v“CrVs, where c:CVvÑCV˜vis the comparison map. 64 CHAPTER 3. A COMBINATION THEOREM If πp Tpxq “ vis not in the support of rVs, let ebe the last edge in the geodesic connecting vto TrVs, so that e`PTrVs. Define (3.8) πrVspxq:“c˝πVe``φe`pXeq˘ĎCV˜v“CrVs, where c:CVe`ÑCV˜vis the comparison map. Lemma 3.2.5. The projections defined in Equation (3.7) and Equation (3.8) are uniform coarsely lipschitz maps. Moreover, they are uniformly coarsely surjective. Proof. In Equation (3.7) the projections are defined as a composition of a uniform quasi isometry with a uniform coarsely lipschitz map. Therefore, it suffices to show that the projections in Equation (3.8) are uniformly coarsely lipschitz too. To prove so, notice that the edge econnects the vertex e´, which lies outside of TrVs, with the vertex e`PTrVs, and notice that there exists a representative Ve`PSe`of rVs. This means that Ve`Ufor any UPSe´, that is Ve`PSe`zφ♦ e`pXeq. As all hieromorphisms are full and coarsely lipschitz, invoking Theorem 2.2.1 we know that the set πVe`pφe`pXeqq are uniformly bounded. Therefore the projections as defined in Equation (3.8) are uniformly coarsely lipschitz, because the comparison maps care uniform quasi-isometries and the sets on which they are applied to are uniformly bounded. These projections are uniformly coarsely surjective, because the projections of the vertex spaces are, following the assumption of Remark 1.6.4. 3.2.3 Projections between hyperbolic spaces Given an equivalence class rVs, define ρrVs p Tto be the support TrVsof the equivalence class rVs, which is uniformly bounded in p Tbecause it is coned-off. Define ρp T rVs:p TÑCrVsas follows. For wPTzTrVs, consider the geodesic connecting wto TrVs, and let ebe its last edge, so that e`PTrVs. Define (3.9) ρp T rVspwq:“c˝πVe``φe`pXeq˘ĎCV˜v“CrVs, where c:CVe`ÑCV˜vis the comparison map. If wPTrVs, then ρp T rVspwqcan be chosen arbitrarily. On the other hand, if wPp TzT, that is wis a cone point, then define ρp T TrVspwq “ ρp T TrVspw1q, where w1is an arbitrarily chosen vertex in the support tree associated to the cone-point w. For an element TrWsPS2, define ρTrWs p Tto be TrWs, and ρp T TrWs:p TÑp TrWsas follows. For vPT, let ρp T TrWspvqbe the closest-point projection (in the tree T) of vonto TrWs. On the other hand, if vPp TzT, that is vis a cone point, then define ρp T TrWspvq “ ρp T TrWspv1q, where v1is any of the points in the support tree associated to the cone-point v. To define the projections ρrVs rWsbetween („-classes of) hyperbolic spaces, we proceed as follows. 3.2. ENDOWING A TREE OF HHS WITH AN HHS STRUCTURE 65 If rVsĎrWsor rVs&rWs, then we define the projections as in [14, Theorem 8.6]. In particular, if rVsĎrWsthere exist vertices v, w, v1such that Vv, Wware the favorite representatives of rVs and rWsrespectively, Vv1and Wv1are representatives of rVsand rWs(possibly different from the favorite ones), and Vv1ĎWv1. Moreover, let cV:CVv1ÑCVvand cW:CWv1ÑCWwbe comparison maps. Define (3.10) ρrVs rWs“cW´ρVv1 Wv1¯ĎCWw“CrWs, which is a uniformly bounded set in CrWs, and define ρrWs rVs:CrWs Ñ CrVsas (3.11) ρrWs rVs“cV˝ρWv1 Vv1˝¯ cW, where ¯ cWis a quasi inverse of cWand ρWv1 Vv1:CWv1ÑCVv1is the projection provided by the hierarchical hyperbolicity of the vertex space pXv1,Sv1q. Analogously, if rVs&rWsand there exists a vertex w1PTsuch that Sw1contains representatives Vw1&Ww1of rVsand rWs, then define (3.12) ρrVs rWs“cW´ρVw1 Ww1¯ĎCWw“CrWs and (3.13) ρrWs rVs“cV´ρWw1 Vw1¯. If there is no common vertex for the supports of rVsand rWs, let v, w be the closest pair of vertices such that Sv,Swcontain representatives Vvof rVsand Wwof rWsrespectively, and let ebe the last edge of the geodesic starting at wand ending at v“e`. Define (3.14) ρrWs rVs“c˝πVe`pφe`pXeqq, where c:CVvÑCV˜vis the comparison map to the favorite representative. In a completely symmetrical way we also define ρrVs rWs. For two elements TrVsand TrV1sof S2, if TrVsĹTrV1sthen define ρTrVs TrV1sto be p TrVs, which is uniformly bounded in p TrV1ssince it is coned-off. Define ρTrV1s TrVs:p TrV1sÑp TrVsas the closest-point projection. If TrVs&TrV1s, then ρTrV1s TrVsand ρTrVs TrV1sare either the closest-point projections (if TrVsand TrV1sdo not intersect), or are defined to be p TrVsXp TrV1s, which by (the proof of) Lemma 3.2.2 is equal to p TrVv_V1 vs, where Vvand V1 vare representatives of rVsand rV1sin a vertex vPTrVsXTrV1s. Notice that if TrVsXTrV1sis not empty, then it is properly contained in both TrVsand TrV1s, and therefore will be coned-off in both p TrVsand p TrV1s. Finally, we define projections between an equivalence class rWsand an element TrVsPS2as 72 CHAPTER 3. A COMBINATION THEOREM Notice that dWe`´πWe`pφe`pXeqq, ρS1 e We`¯—dWe``πWe`pφe`pXeqq, πWe`pFS1 eq˘ ďKdpφe`pXeq,FS1 eq`K, and so, by Theorem 2.3.3, we have that (3.21) dWe`´πWe`pφe`pXeqq, ρS1 e We`¯ďKη `K. Combining Equation (3.20) and Equation (3.21) we obtain that drWs`ρrVs rWs, ρrUs rWs˘is uniformly bounded. Assume now that TrUsXTrWs“ H: in particular rUs&rWs. By Lemma 3.1.2 we know that TrVsĎTrUs. Therefore, there exists an edge eseparating TrVs(and TrUs) from TrWs, so that e`PTrWs. As defined in Equation (3.14), we have that ρrVs rWs“cW˝πWe`pφe`pXeqq “ ρrUs rWs. Therefore ρrVs rWs“ρrUs rWs, and drWspρrUs rWs, ρrVs rWsq “ 0 is uniformly bounded. TrW1s&TrW2sLet TrW1s, TrW2sPS2satisfying TrW1s&TrW2s, and let xPX. In this case, we always have that min dTrW1spπTrW1spxq, ρTrW2s TrW1sq, dTrW2spπTrW2spxq, ρTrW1s TrW2sq(“0, because ρTrW1s TrW2sand ρTrW2s TrW1sare defined as closest-point projections if TrW1sXTrW2s“ H, or as the (coned-off) intersection, if it is non-empty. TrW1sĎTrW2sLet TrW1s, TrW2sPS2satisfying TrW1sĎTrW2s. Consistency follows, because for all xPXwe have that πTrW1spxq “ ρTrW2s TrW1s`πTrW2spxq˘. Therefore diamCTrW1s`πTrW1spxqYρTrW2s TrW1spπTrW1spxqq˘“0, where CTrVs“p TrVs, and the consistency inequality is satisfied. Let TrW3sPS2be such that either 1. TrW1sĎTrW2sĹTrW3s, or 2. TrW2s&TrW3s. In either case we have that ρTrW1s TrW3sĎρTrW2s TrW3s, and therefore dTrW3s`ρTrW1s TrW3s, ρTrW2s TrW1s˘“0. Let now rVs P S1be such that rVs&TrW2sand rVs M TrW1s. We want to prove that drVspρTrW1s rVs, ρTrW2s rVsq is uniformly bounded. We now check every possible case. If the support of rVsdoes not intersect TrW2s(and therefore, does not intersect TrW1sĎTrW2s), then ρTrW1s rVs“ρTrW2s rVsand the claim is satisfied. If the support TrVsintersects both TrW1sand TrW2s, then also in this case we have that ρTrW1s rVs“ρTrW2s rVs. Finally, if TrVsintersects TrW2sbut not TrW1s, then ρTrW1s rVs“c˝πVe``φe`pXeq˘, 3.2. ENDOWING A TREE OF HHS WITH AN HHS STRUCTURE 73 where eis the last edge in the geodesic connecting TrW1sto TrVs, the vertex e`lies in TrVs, and Ve`is the representative of rVsin Se`. On the other hand, ρTrW2s rVs“ρrW2s rVs, and rW2s&rVs. As both classes rVsand rW2sare supported on the vertex e`, we have that ρrW2s rVs“c˝ρW2e` Ve`, where W2e`is the representative of rW2sin that vertex. By Lemma 2.3.4 we have that πVe``φe`pXeq˘is coarsely equal to ρr Se` Ve`, where r Se`“φ♦ e`pSeqand Seis the Ď-maximal element of Se. Therefore drVspρTrW1s rVs, ρTrW2s rVsqis uniformly bounded. rVs&TrWsLet TrWsPS2. If TrVsXTrWs“ H, then min drVspπrVspxq, ρTrWs rVsq, dTrWspπTrWspxq, ρrVs TrWsq(“0,@xPX. Thus, suppose that the intersection is non-empty. Since rVs&TrWsit follows that rVs&rWs. Suppose that dTrWs`πTrWspxq, ρrVs TrWs˘is big, so that in particular xRTrVsXTrWs“ρrVs TrWsand the geodesic connecting xto TrVspasses through the set TrWs. By definition, πrVspxq “ c˝πVe``φe`pXeq˘, and ρTrWs rVs“ρrWs rVs“cpρWe` Ve`q, where e`is the vertex of the edge ethat belongs to TrVsXTrWs, while e´PTrWszTrVs, and Ve`and We`are the representatives of rVsand rWsrespectively at the vertex e`. Let Sebe the Ď-maximal element of Se. As the equivalence class rVsis not supported in the vertex e´, it follows that Ve`is not nested into φ♦ e`pSeq “ r Se. On the other hand We`Ďr Se. Therefore, ρWe` Ve`and ρr Se Ve`coarsely coincide by Definition 1.6.1(4), and by Lemma 2.3.4 we obtain that πVe``φe`pXeq˘—ρ˜ Se Ve`—ρWe` Ve`, that is, πrVspxqand ρTrWs rVscoarsely coincide. Thus, drVs`πrVspxq, ρTrWs rVs˘is uniformly bounded. rVsĎTrWsIf the distance dTrWspπTrWspxq, ρrVs TrWsq ą κ0, it follows in particular that πTpxq R ρrVs TrWs“TrVsXTrWs, and that the geodesic in p Tconnecting xto ρrVs TrWspasses through the set TrWszTrVs. In this case, we have that πrVspxq “ πrVs`πTrWspxq˘is equal to ρTrWs rVs`πTrWspxq˘. Therefore the consistency inequality is satisfied also in this case. (Finite complexity) It is enough to show finite complexity in S1and S2independently. Finite complexity in S1follows from [14, Lemma 8.11]. For S2, notice that any chain of proper nestings TrU1sĽTrU2sĽ¨¨¨ ĽTrUns induces the chain of proper nestings rU1sĹrU2sĹ. . . ĹrUnsin S1, by Corollary 3.2.3. As only equivalence classes are allowed to be nested into an intersection of supports, and not vice versa, finite complexity is proved. In particular, it follows that the complexity of `XpTq,S˘is twice the complexity of S1plus one, and the complexity of S1is maxvχv`1, where χvis the complexity of the vertex space pXv,Svq. (Large links) Let rWs P S1and x, x1PX. Suppose that xPXvand x1PXv1for some v, v1PT, and let wbe the favorite vertex for rWs. Let Edenote the maximal of the constants Evof the Bounded Geodesic Axiom of the hierarchically hyperbolic space pXv,Svq. 74 CHAPTER 3. A COMBINATION THEOREM Suppose that, for some rVsĎrWs, we have drVspπrVspxq, πrVspx1qq ě E1, where E1depends on E and on the quasi-isometry constants of the edge hieromorphisms. Then dVwpc˝πVvpxq,c˝πVv1px1qq ě E, for a representative VwPSwof rVs. As the large links axiom holds in Sw, we have that VwĎTi, where tTiPSwuN i“1is a set of Nelements in Sw, where N“tdrWspπrWspxq, πrWspx1qquand each Tisatisfies TiĹWw. Moreover, the Large Links Axiom in Swimplies that drWspπrWspxq, ρrTis rWsq “ dWwpcW˝πWvpxqq, ρTi Wwq ď Nfor all i“1, . . . , N. Thus the large links axiom for elements rVs P S1 and rUs P SrVsfollows. We now consider the case of TrWsPS2, and XPSTrWs. This can happen both when Xis an equivalence class, or when XPS2. We deal with the case XPS2in the following lemma, whilst the case X“ rVs P S1is considered after the lemma. Lemma 3.2.7. Let x, x1PXand SPS2Ytp Tu. The set Y“ tXPS2|XĹS, dXpπXpxq, πXpx1qq ą 4u is finite. Moreover, the set of Ď-maximal elements in Yhas cardinality bounded linearly in terms of the distance dS`πSpxq, πSpx1q˘. Proof. Let σbe the geodesic in Tconnecting v“πTpxqto v1“πTpx1q. We begin by noticing that, if XXσ“ H, then dXpπXpxq, πXpx1qq “ 0 because these two sets coincide, and therefore XRY. In particular, as nesting between elements of tp TuYS2is inclusion, if σdoes not intersect Sthen Ywill be empty, and the lemma is trivially satisfied. Suppose now that σintersects S, and consider the map ϕ:YÑPpσqdefined as ϕpXq “ XXσ, where Ppσqis the set of subpaths of σ. We first prove that ϕis an injective map. Let X, X1PY be such that X‰X1and, looking for a contradiction, suppose that ϕpXq “ ϕpX1q, so that XXσ“X1Xσand therefore XXσ“XXX1Xσ. Since Xintersects σ, we have that πXpxqand πXpx1qare vertices of σ. Therefore πXpxqand πXpx1q lie in XXσĂXXX1. Since XXX1is properly contained in both Xand X1, it will be coned-off in both CXand CX1by construction. Therefore dXpπXpxq, πXpx1qq ď 2, which contradicts the definition of the set Y. Therefore the map ϕis injective, and the set Yis finite. We now claim that, for elements X, X1PY, we have that ϕpXq Ĺ ϕpX1qif and only if XĹX1. Indeed, if XĹX1, that is XĹX1, then ϕpXq Ĺ ϕpX1q. On the other hand, suppose that ϕpXq Ĺ ϕpX1q, and let X“TrVsand X1“TrV1s, for some equivalence classes rVsand rV1s. Since ϕpXq “ XXσĹϕpX1q “ X1Xσ, we have that (3.22) XXσ“XXX1Xσ. 3.2. ENDOWING A TREE OF HHS WITH AN HHS STRUCTURE 75 Moreover, as XXX1“TrVsXTrV1s“TrV_V1s, from Equation (3.22) we obtain that (3.23) TrVsXσ“TrV_V1sXσ. As rVsĎrV_V1s, Lemma 3.1.2 implies that TrV_V1sĎTrVs. If TrV_V1sis properly nested into TrVs, then TrV_V1sis coned off in CTrVs“p TrVs. Equation (3.23) implies that dTrVspπTrVspxq, πTrVspyqq “ 2, which is a contradiction since TrVsPYby hypothesis. Therefore, TrV_V1s“TrVs, which implies that TrVsĎTrV1s, as desired. We now show that Ymax “ tX1, . . . , Xnu Ď Y, the set of Ď-maximal elements in Y, has cardinality at most dSpπSpxq, πSpx1qq. Since every element of Ymax is properly nested into S, it follows that its support is coned off in CS“p S. We now prove that XjXσĘ pXk1Y¨¨¨YXkrqXσfor any pairwise distinct elements Xj, Xk1, . . . , Xkrall belonging to Ymax. The claim was just proved for r“1. Indeed, if XjXσĎXk1Xσthen XjĎXk1, and this contradicts the fact that Xjand Xk1are distinct Ď-maximal elements of Y. Suppose that XjXσĎ pXk1YXk2qXσ, and let TrUjs, TrUk1sand TrUk2sdenote Xj, Xk1and Xk2respectively. In this case, there exists a path in CXjfrom πXjpxqto πXjpx1qthat passes through the cone points of TrUj_Uk1s and TrUj_Uk2s, which are properly nested into Xj. Then, dXjpπXjpxq, πXjpx1qq ď 4, contradicting the assumption that XjPYmax. On the other hand, assume that XjXσĎ pXk1YXk2Y. . . YXkrqXσwhere rą2, ki‰jfor all i,ka‰kbfor all a‰b, and there does not exist ki‰kjsuch that XjXσĎ pXkiYXkjqXσ. We claim that there exists ssuch that XksXσĎXjXσ. Indeed, assume without loss of generality that the endpoints of XjXσare contained in Xk1Xσ and XkrXσrespectively. By hypothesis, XjXσcannot be entirely contained in pXk1YXkrqXσ. Therefore, there exists vPXjXσzpXk1YXkrqXσ, that is vPXksXσfor 1 ăsăr. Note that XksXσcannot contain either of the endpoints of XjXσ, since that would imply that XjXσ is contained in either pXk1YXksq X σor pXkrYXksq X σ. As a consequence we obtain that XksXσĎXjXσ, which is a contradiction, since Xksis maximal with respect to nesting. From here we can conclude that |Ymax|ďdSpπSpxq, πSpx1qq. Indeed, given any Ď-maximal element XiPYmax and its cone point vi, the following dichotomy holds: either viis a vertex in the geodesic path pσ, or not, where pσis a geodesic path in CSconnecting πSpxqto πSpx1q. In the latter case, it must be that pσcontains either one or two edges of the support Xi. Therefore, the bound is proved. Therefore, if dTrUspπTrUspxq, πTrUspx1qq ą 4 for some TrUsPSSztSu, that is TrUsPY, then TrUsĎX for some Ď-maximal element Xof the set Y. We now address the case when Xis an equivalence classes X“ rVs P STrWs. By definition, rVsĎTrWsif and only if rVsis orthogonal to rWs. In particular, it follows that TrVsXTrWs‰ H. 76 CHAPTER 3. A COMBINATION THEOREM If TrVsdoes not intersect the geodesic σthen the distance drVs`πrVspxq, πrVspx1q˘is equal to zero by Equation (3.8), because the edge eappearing in the cited equation will be the same for both x and x1. Now assume that TrVsXσ‰ H. As a fist sub-case, suppose that σXTrWsis empty, let (3.24) I:“ rVsĎTrWs|TrVsXσ‰ H(, and notice that Icould be infinite. Consider the geodesic αconnecting TrWsto σin the tree T, and notice that αhas at least one edge, being TrWsand σdisjoint. For rVs P I, we have that TrVs intersects both TrWsand σ, and therefore αis contained in TrVs, being Tis a tree. Thus the set TrWsXŞrVsPITrVsis not empty, because (at least) the initial vertex on the geodesic αbelongs to this intersection. Let the set Iindex I, that is I“ trVisuiPI. Without loss of generality, we can suppose that each Vi is the representative of rVisin the vertex space pXv,Svq. Let SvPSvbe the Ď-maximal element, and notice that rVisĎrSvsfor all iPI. Furthermore, note that rVsĎrŽiPIVisfor all rVs P Iand let rV_sdenote rŽiPIVis. Therefore, in this first sub-case, Large Links is satisfied by the family YYtrV_su for the elements TrWsPSand x, x1PX. For the second sub-case, suppose that σXTrWsis not empty, and let tv1, . . . , vnube the finitely many vertices of σXTrWs(there can be only finitely many such vertices because σis a geodesic). Analogously to Equation (3.24), for all viPσXTrWsdefine Ivi“ rVsĎTrWs|viPTrVsXσ(, and notice that I“ŤIvi. As in the previous case, for each Iviconsider rSvis, and notice that rVsĎrSvisfor all rVs P Ivi, for all i“1, . . . , n. Therefore, Large Links for an element TrWsPS2 is satisfied considering the set YYtrVv1 _s,...,rVvn _su. Notice that, in both sub-cases, we bounded the cardinality of the sets YY trV_su and YY trVv1 _s,...,rVvn _su in terms of σ, that is in terms of dTpx, x1q. As dTrWspπTrWspxq, πTrWspx1qq is bounded from above by dTpx, x1q, we obtained the desired bound on the cardinality of these sets. Combining these bounds with Lemma 3.2.7, we conclude the proof of Large Links for the case XĹTrWs. Finally, we prove Large Links for the Ď-maximal element p T. From Lemma 3.2.7 applied with S“p T, there are only finitely many (and the number depends only on the distance in p Tfrom xto x1) elements XPS2such that dXpπXpxq, πXpx1qq is big. On the other hand, for an equivalence class rVsĎp T, the distance drVspπrVspxq, πrVspx1qq can be big only if the support TrVsintersects the geodesic σconnecting vto v1(otherwise, it would be zero). Let S1, . . . , Snbe the Ď-maximal elements of all the finitely many edges in σXTrVs. We have that rVsĎrSisfor all i“1, . . . , n. Therefore, the set YYtS1, . . . , Snuis the set of significant elements for the Axiom. Let E1be the constant that satisfies the Large Links Axiom of the (uniformly) hierarchically hyperbolic vertex spaces (see Definition 1.6.1), and let Eąmaxt2, E1u. Then Large Links is satisfied with this constant E. 3.2. ENDOWING A TREE OF HHS WITH AN HHS STRUCTURE 77 (Bounded geodesic image) Consider rWsĹp T, and let γbe a geodesic in ˆ T. If γXTrVs“ H, let ebe the last edge in the geodesic connecting γto TrVs, and suppose e`PTrVs. Then ρˆ T rVspγq “ cW˝πVe`pφe`pXeqq is a uniformly bounded set. If not, then γintersects ρrVs p T. The cases rVsĎTrW1s,TrW1sĎTrW2s, and TrW1sĎˆ T, where TrW1s, TrW2sPS2, are analogous. Let rWs P S, let rVsĎrWs, and let γbe a geodesic in CrWs “ CWw(where wis the favorite vertex of rWsand WwPSwis the favorite representative). Let Vwbe the representative of rVs supported in the vertex w, so that ρrVs rWs“ρVw Ww. The Bounded Geodesic Image Axiom in this case follows because it holds in the vertex space pXw,Swq(notice that the constant Echanges according to the quasi-isometry constant of the comparison maps). (Partial realization) Notice that two elements TrW1sand TrW2sof S2are never orthogonal. Consider k`1 pairwise orthogonal elements rV1s,...,rVks, TrWsPS, and let piPπrVispXq Ď CrVis, for i“1, . . . , k, and vSPp TrWs. By definition of orthogonality, TrVisXTrVjs‰ H for all i‰j,TrWsĎTrVisfor all i“1, . . . , n, and in particular TrWsĎŞk i“1TrVis. Consider a vertex vPTrWsthat is not a cone point and has distance at most one from vS, that is vPTXTrWsand dTrWspv, vSq ď 1. As vPTrVis for all i“1, . . . , k, without loss of generality we can suppose that Viis an element of Sv, by choosing representatives. We have that ViKVjfor all i‰j. Comparison maps are uniform quasiisometries, and piPπrVispXq, therefore the element cippiqis uniformly close to the set πVipXqfor all i“1, . . . , k, where ci:CrVis Ñ CViis the comparison map. For i“1, . . . , k, let pv iPπVipXqbe a point such that dVi`pv i,cippiq˘is uniformly bounded. By Partial realization in the vertex space pXv,Svq, there exists xPXvsuch that dVipπVipxq, pv iq is uniformly bounded for all i. As comparison maps are uniform quasi-isometries, we obtain that drVispπrVispxq, piqis uniformly bounded for all i. Moreover, dTrWspπTrWspxq, vSq “ dTrWspv, vSq ď 1. If rVisĎrUs, then rUshas a representative UvPSvsuch that ViĎUv. Therefore drUspπrUspxq, ρrVis rUsq is uniformly bounded, because xis a realization point for tViuk i“1, and comparison maps are uniform quasi isometries. If rVisĎTrUs, then ρrVis TrUs“TrVisXTrUsand πTrUspxq P ρrVis TrUs. Therefore, dTrUs`πTrUspxq, ρrVis TrUs˘“ 0. Analogously, for TrWsĎTrUswe have that dTrUs`πTrUspxq, ρTrWs TrUs˘“0. This argument also applies when considering the Ď-maximal element, therefore proving that dp T`πp Tpxq, ρTrWs p T˘“0 and dp T`πp Tpxq, ρrVis p T˘“0. Let now rVis&rUs. Either TrUsXTrVis“ H, in which case the distance drUspπrUspxq, ρrVis rUsqis uniformly bounded, or TrUsXTrVis‰ H, in which case rUshas a representative UvPSvthat is transverse to Vi. Therefore, in the latter case the distance drUspπrUspxq, ρrVis rUsqis again uniformly bounded, because it is in the vertex space Xv, and comparison maps are uniform quasi-isometries. If rVis&TrUsthen πTrUspxq P ρrVis TrUs, and therefore dTrUs`πTrUspxq, ρrVis TrUs˘“0. For the last case, suppose that TrWs&rUsfor some rUs P S1. If the support of rUsdoes not intersect TrWs, then πrUspxq P ρTrWs rUs. So, suppose that TrWsintersects TrUs. Again using Lemma 2.3.4, we can conclude. If TrWs&TrUsand TrWsXTrUs‰ H, then the subtree TrWsXTrUs“TrW_Usis strictly contained in TrUs. Therefore, TrWsXTrUsis coned-off in CTrUs“p TrUs. Since πTrUspxq P TrWsXTrUs, we obtain that dTrUspπTrUspxq, ρTrWs TrUsq ď 2. On the other hand, if TrWsXTrUs“ H then πTrUspxq “ 78 CHAPTER 3. A COMBINATION THEOREM ρTrWs TrUs“e`PTrUs, where eis the last edge in the geodesic separating TrWsfrom TrUs, and therefore dTrUspπTrUspxq, ρTrWs TrUsq “ 0. By definition, no element of S1can be nested into an element of S2. Therefore, all the relevant cases have been considered. (Uniqueness) Suppose x, y PXare such that dRpπRpxq, πRpyqq ď K, for all RPS. In particular, we have that dp T`πp Tpxq, π p Tpyq˘ďK, that dS`πSpxq, πSpyq˘ďKfor all SPS2, and that drVs`πrVspxq, πrVspyq˘ďKfor all rVs P S1. Suppose that the distance in p Tfrom πp Tpxqto πp Tpyqis realized by a path only consisting of vertices of TĎp T, and let v0“πTpxq, v1, . . . , vk´1, πTpyq “ vk, be these vertices, where kďK. In particular, no four consecutive vertices can belong to the same support tree, because this would produced a shorter path in p Tjoining xto y. We have that dXpx, yq ď řk i“0dXvi`gvipxq,gvipyq˘`k. Moreover, for all i“0, . . . , k we have that the distance dXvi`gvipxq,gvipyq˘is uniformly bounded. Indeed, if this is not the case, by Uniqueness in the hierarchically hyperbolic space pXvi,Sviq, there exists VPSvisuch that dV`πVpgvipxqq, πVpgvipyqq˘is not bounded. By [14, Lemma 8.18] and Theorem 2.2.1, we have that dV`πVpgvipxqq, πVpgvipyqq˘and drVs`πrVspxq, πrVspyq˘coarsely coincide, and therefore the latter is not bounded. This contradicts the fact that drVs`πrVspxq, πrVspyq˘ďK, and thus dXvi`gvipxq,gvipyq˘ďζ“ζpKqis uniformly bounded, as claimed. Therefore, dXpx, yq ď ζ1pKq, for some uniform bound ζ1pKq. Suppose now that in the geodesic σin ˆ Tconnecting πˆ Tpxqto πˆ Tpyqthere is a cone point. Therefore, there exists an element TrW1sPS2containing two points x1and y1in this geodesic (that, therefore, have distance two in ˆ Tsince TrWsis coned-off in p T). As TrW1sPS2, we have that dTrW1s`πTrW1spx1q, πTrW1spy1q˘“dTrW1spx1, y1q ď K. Either the geodesic σ1in CTrW1s“p TrW1s connecting these two points only consists of vertices of T, or there are cone points, and therefore an element TrW2sPS2containing two elements x2, y2of the geodesic σ1. As complexity in S2is finite and nesting coincides with inclusion, this process must end after a finite number of steps (that depends only on K). Therefore, there exists a geodesic in Tconnecting πˆ Tpxqto πˆ Tpyq, whose length is bounded from above by a function in K. Repeating the argument given before, we conclude that dXpx, yqis uniformly bounded. This concludes the proof of hierarchical hyperbolicity of the space `XpTq,S˘. 3.3 Applications Theorem 3.0.1 has two main applications. The first one is a combination theorem on hierarchically hyperbolic groups (Corollary 3.3.1). The second one is for graph products of hierarchically hyperbolic groups (Theorem 3.3.7). We now show their proofs. 3.3. APPLICATIONS 79 3.3.1 Graph of hierarchically hyperbolic groups Corollary 3.3.1. Let G“`Γ,tGvuvPV,tGeuePE,tφe˘:GeÑGe˘uePE˘be a finite graph of hierarchically hyperbolic groups. Suppose that: 1. each edge-hieromorphism is hierarchically quasiconvex, uniformly coarsely lipschitz and full; 2. comparison maps are isometries; 3. the hierarchically hyperbolic spaces of Ghave the intersection property and clean containers. Then the group associated to Gis itself a hierarchically hyperbolic group. We begin with the following lemma, in which we use the notation of Section 3.1.1. Lemma 3.3.2. Let Tbe a tree of hierarchically hyperbolic spaces and r Tbe the corresponding decorated tree. Then 1. for every support tree TrVsPS2πr TrVs‹pXpTqq is isometric to CTrVs, and quasi-isometric to Cr TrVs‹, for all support trees ; 2. πrVs‹pXpTqq is isometric to πrVspXpTqq, and quasi-isometric to πrVs‹pXpr Tqq, for all equivalence classes rVs P S1; 3. XpTqis hierarchically quasiconvex in Xpr Tq. Proof. 1. The first assertion of this item follows from the fact that the projections to hyperbolic spaces for elements in XpTqare not modified by decorating the tree T. Furthermore, by the construction of Section 3.1.1, there exists a constant Cą0 such that Cr TrVs‹“NC`πr TrVs‹pXpTqq˘, and therefore πr TrVs‹pXpTqq is quasi-isometric to Cr TrVs‹. 2. As the favorite representative of the equivalence class rVs‹is the same as of the class rVs, it follows that πrVs‹pXpTqq is isometric to πrVspXpTqq. The second assertion of this item follows from the equality Xpr Tq “ NC`XpTq˘. 3. By what was just proved in the previous points, πUpXpTqq is kp0q-quasiconvex in πUpXpr Tqq, for all UPS, for some fixed number kp0q. Moreover, let ~ bbe a κ-consistent tuple such that bXPπXpXpTqq for every XPSand let xPXpr Tqbe a realization point of ~ b. Since Xpr Tq “ NCpXpTqq there exists x1PXpTqsuch that dXpr Tqpx, x1q ď C, and therefore the proof is complete. As already mentioned in Section 3.1.1, to construct the hierarchically hyperbolic structure of the graph of hierarchically hyperbolic groups Gof Corollary 3.3.1, we do not consider directly a 80 CHAPTER 3. A COMBINATION THEOREM decorated tree, because there might not be a non-trivial action of the fundamental group of Gon that hierarchically hyperbolic space. Instead, we proceed as follows. Let (3.25) T“´T, tHwuwPV,tHfufPE,tφf˘u¯ be the tree of hierarchically hyperbolic groups associated to G, as described in [14, Section 8.2]. In particular, T“ pV, Eqis the Bass-Serre tree associated to the finite graph Γ, each Hwis conjugated in the total group Gto Gv, where wmaps to vvia the quotient map TÑΓ, analogously Hfis conjugated to Ge, and the edge maps φf˘agree with these conjugations of edge and vertex groups to give the embeddings in the tree of hierarchically hyperbolic groups. Let XpTqbe the associated metric space, and let Sdenote the index set associated to XpTq, as described in Section 3.2. Associated to this, we consider the decorated tree r Tof hierarchically hyperbolic groups, as described in Section 3.1.1. By Theorem 3.0.1, the metric space Xpr Tqadmits a hierarchically hyperbolic space structure, that we denote by r S. By Lemma 2.1.15, the metric space XpTqis hierarchically quasiconvex in Xpr Tq, and therefore `XpTq,r S˘is a hierarchically hyperbolic space by [14, Proposition 5.5], where the hyperbolic spaces associated to an element UPr Sis defined as πU`XpTq˘ĎCU. From Remark 1.6.4, we are assuming that every πUis uniformly coarsely surjective, so in fact there is no harm in considering CUinstead of πU`XpTq˘. As r Sand Scoincide as sets of indices (what changes are the hyperbolic spaces associated to each index, as detailed in Section 3.1.1), the above substitution is equivalent to equipping the metric space XpTqwith the hierarchically hyperbolic structure given by S. That is to say, `XpTq,S˘is a hierarchically hyperbolic space. We now set to prove Corollary 3.3.1. Before showing the full proof we discuss how the index set constructed in Section 3.2 on a tree of hierarchically hyperbolic spaces can be applied to the hierarchical hyperbolic group structure of a graph of groups Gon the tree of spaces obtained by considering its Bass-Serre tree. We first describe the hierarchical hyperbolic space structures involved in each vertex space associated to the tree of spaces described in Equation (3.25). Remark 3.3.3. Recall that each vertex in the Bass-Serre tree Tcorresponds to a coset gGv, where Gvis a vertex group corresponding to the graph of groups G. We endow the metric space gGvwith a copy of the index set Svdenoted by gSvsuch that there is a hieromorphism φg:pGv,Svq Ñ pgGv, gSvqequivariant with respect to the conjugation isomorphism GvÑGg v. If UPSvwe denote by φpUq gthe isometry at hyperbolic space level making the following diagram commute: Gv φg// πVv  gGv πgVv  CVvφpVvq g //CgVv We recall here the notion of T-coherent bijections, where Tis the tree of hierarchically hyperbolic 3.3. APPLICATIONS 81 spaces. A bijection of the index set Sgiven in Equation (3.4) is said to be T-coherent if: •it induces bijections on the sets S1and S2; •it preserves the relation „on S1; •it induces a bijection bof the underlying tree Tthat commutes with f:ŮvPVSvÑT, where fsends each VPSvto the vertex v. That is, fb “bf. Notice that the composition of T-coherent bijections is T-coherent. Therefore, let PTďAutpSq be the group of T-coherent bijections. To produce the index set Sin a PT-equivariant manner, we proceed as follows. Notice that Gacts on ŮvPVSv, so that for any VvPSvwe have that g.VvPSg.v. This extends to an action of S1 defining g.rVs “ rg.V s. For any rWs P S1, choose a left transversal SrWsof the subgroup StabGprWsq “ gPG|grWs“rWs(, and impose that eGPSrWs. For each PT-orbit in S1choose a representative rVsof the orbit, a favorite vertex vfor rVs, and a favorite representative VvPSvfor rVs. For any element gPG, there is a unique element lPSrVssuch that gPl¨StabGprVsq. We declare lv to be the favorite vertex of grVs, and gVvPSl.v to be the favorite representative of the equivalence class g.rVs. This definition is consistent, that is that if g, ˜gPG, then the favorite vertex of pg˜gq.rVscoincides with the favorite vertex of g.`˜g.rVs˘. Indeed, suppose that ˜gP˜ l¨StabGprVsq, that g˜gPp¨ StabGprVsq, and that gPl‹¨StabGp˜grVsq, for unique elements ˜ l, p PSrVsand l‹PS˜grVs. Thus, the favorite vertex of g˜grVsis p.v, and its representative is Vp.v PSp.v. On the other hand, the favorite vertex of ˜grVsis ˜ l.v, with favorite representative ˜ lrVs, and consequently the favorite vertex of g`˜grVs˘is pl‹˜ lq.v, with favorite representative Vpl‹˜ lq.v. As gPl‹¨StabGp˜grVsq and StabGp˜grVsq “ ˜gStabGprVsq˜g´1, we have that g˜gP pl‹˜gq¨StabGprVsq “ pl‹˜ lq¨StabGprVsq. Therefore, as g˜gbelongs to a unique coset of StabGprVsq, we have that p¨StabGprVsq “ pl‹˜ lq¨StabGprVsq, which implies that l‹˜ lrVs “ pp´1l‹˜ lrVs “ prVs. As a consequence, the favourite vertices and representatives of grgrVsand gprgrVsq are equal. From the definition of the action of PTon S2, it follows that Cg.TrUs“CTg.rUs. Lemma 3.3.4. Let G“`Γ,tGvuvPV,tGeuePE,tφe˘:GeÑGe˘uePE˘be a finite graph of hierarchically hyperbolic groups satisfying the hypotheses of Corollary 3.3.1. Further, let Tbe the tree of hierarchically hyperbolic spaces associated to Gas in Equation (3.25). If gPG“π1pGqsuch that grVs“rWsthen for every rvPTrVsand representative VrvPSrvof rVsthere exist an isometry gVrv:CVrvÑCWg.rvmaking the following diagram uniformly coarsely commute Grv g// πVrv  Ggrv πWg.rv  CVrvgVrv //CWg.rv 88 CHAPTER 3. A COMBINATION THEOREM For each rVs P S1, the projection πrVs, as defined in Equation (3.7) and Equation (3.8), is πrVspxq “ $ ’ & ’ % cw˝πVwpxq,@xPXv, v PTrVs; ce`˝πVe`pφe`pXeqq,@xPXv, v RTrVs, where e“epvqis the last edge in the geodesic connecting vto TrVssuch that e`PTrVs, and the maps cwand ce`denote the appropriate comparison maps to the favorite representative of rVs. Let xPXvĎXand let TrVsPS2. Then, πTrVspxqis defined as the composition of the closest point projection of vto TrVsin the Bass-Serre tree T, with the inclusion of TrVsinto the conedoff CTrVs“p TrVs. To prove that GΓztvuis hierarchically quasiconvex in GΓ, we need to check the two conditions of Definition 1.7.4. For each element TrVsPS2we have that πTrVspGΓztvuqis a point in CTrVs“p TrVs and, therefore, it is quasiconvex in CTrVs. Suppose that rVs P S1, and assume that rVshas a representative in g.Sv, where Svis the index set associated to the vertex group Gv. In particular rVs “ tVu, and πrVspGΓztvuq Ď πVpg.Glinkpvqq. Since VRg.Slinkpvq, the set πVpg.Glinkpvqqis uniformly bounded, and therefore πrVspGΓztvuqis quasiconvex in CrVs. On the other hand, assume that the group orbit G.rVsintersects SΓztvu. Without loss of generality, as the group acts isometrically on the hyperbolic spaces, we can assume that rVshas a representative ˜ VPSΓztvu. By definition πrVspGΓztvuq “ c˝π˜ VpGΓztvuq, where cis the comparison map from ˜ Vto the favourite representative of rVs. By Axiom (1) of Definition 1.6.1, the set π˜ VpGΓztvuq is quasiconvex in C˜ V, and therefore πrVspGΓztvuqis quasiconvex in CrVs, being can isometry. It follows that for every element rVs P S1, the set πrVspGΓztvuqis quasiconvex in CrVs. To conclude the proof of hierarchical quasiconvexity, consider a consistent tuple ~ bin pG, Sqsuch that brVsPπrVspGΓztvuqand bTrVsPπTrVspGΓztvuqfor every rVs P S1. The sets πTrVspGΓztvuqare uniformly bounded, being points, for all TrVsPS2. Moreover, πrVspGΓztvuqare uniformly bounded for every equivalence class rVs P S1which has a representative in g.Sv. Let αdenote the vertex of the Bass-Serre tree in which the subgroup GΓztvuis supported. Let i:GΓztvuÑGΓbe the hieromorphism defined as follows. At the metric-space level define it to be the natural inclusion. At the level of index sets i♦pUq“rUsand, at the level of hyperbolic spaces, i˚ U:CUÑCrUsis the comparison map c:CUαÑCrUs, which is an isometry. For each rVs P S1, we have that πrVspGΓztvuq “ $ ’ & ’ % cα˝πVαpGΓztvuq,if αPTrVs; ce`˝πVe`pφe`pXeqq,if αRTrVs. 3.3. APPLICATIONS 89 By Theorem 2.2.1 the set πVe`pφe`pXeqq is uniformly bounded, and thus ce`˝πVe`pφe`pXeqq is uniformly bounded. For each rVs P S1such that αPTrVs, let crVsdenote cpbrVsq, where the maps cdenote the comparison maps (which are isometries) from the favourite representative of rVsto the representative Vα(therefore, the maps cchange with respect to different equivalence classes). Consider the consistent tuple ~c “ź rVsPS1, αPTrVs crVs By induction hypothesis, GΓztvuis a hierarchically hyperbolic group. Therefore, the consistent tuple ~c admits a realization point zPGΓztvu, and thus we obtain that πrVspzq — brVsfor every rVs P S1. Furthermore, since πTrVspGΓztvuqis a point, we also have that πTrVspzq “ bTrVs“πTrVspGΓztvuqfor every TrVsPS2. That is, the second condition of hierarchical quasiconvexity is proved, and the inclusion GΓztvuãÑGΓis a hierarchically quasiconvex hieromorphism. Moreover, for each VPSΓztvuthe map CVÑCrVsis an isometry. Note that, if an element rVsĎi♦pUq“rUs, where UPSΓztvu, then TrUsĎTrVs. By assumption αPTrUs, and therefore αPTrVsand there exists VPSΓztvusuch that i♦pVq “ rVs. Thus, we proved that all induction hypotheses are satisfied by the inclusion GΓztvuãÑG, that is that the embedding is a full, hierarchically quasiconvex hieromorphism, which induces isometries at the level of hyperbolic spaces. To deduce the same for an arbitrary G∆, we proceed as follows. If ∆ “Γztuufor some (other) vertex uPV, then the above argument, where in Equation (3.29) we consider the splitting over the subgroup Glinkpuq, proves that the inclusion G∆ãÑGsatisfies the desired properties. If not, then ∆ is a proper subgraph of Γztuu, for some uPV. Induction proves that the embedding G∆ãÑGΓztuusatisfies said properties, and again the above argument proves the claim for the inclusion GΓztuuãÑG. As fullness, hierarchical quasiconvexity, and inducing isometries at the level of hyperbolic spaces, are all properties preserved by composition of hieromorphisms, we conclude that the inclusion G∆ãÑGsatisfies the inductive statement, and the proof is thus complete. We end the chapter with a remark that anticipates what the following chapter is about. In short, it shows the limits of application of Theorem 3.0.1 to general graphs of groups. Example 3.3.8. [Baumslag–Solitar groups] Let us consider more in detail non-euclidean Baumslag– Solitar groups BSp1, kq “ xa, t |tat´1“aky, where k‰ ˘1. Let T“ pV, Eqbe the Bass–Serre tree associated to the HNN extension BSp1, kq, so that V“ tgxay | gPBSp1, kqu. Two distinct vertices gxayand hxayare joined by an edge ePEif and only if there exists bP xaysuch that either hxay “ gbt˘1xay, or hxay “ gbt´1xay. For a vertex gxay “ vPVlet `Xv,Sv˘:“`gxay,txayu˘ be the hierarchically hyperbolic space associated to the vertex, and for any edge ePElet `Xe,Se˘:“`xay,txayu˘be the hierarchically hyperbolic space associated to the edge. Given 90 CHAPTER 3. A COMBINATION THEOREM tgxay, hxayu “ ePE, consider the hieromorphisms φe`:`xay,txayu˘Ñ`gxay,txayu˘be defined as φe`paq “ ga, and φe´:`xay,txayu˘Ñ`hxay,txayu˘be defined as φe´paq “ hak. We have that T“´T, ``Xgxay,Sgxay˘˘(gPG, `Xe,Se˘u˘(ePE, φe˘(ePE¯ is a tree of hierarchically hyperbolic spaces. The vertex-spaces and edge-spaces all have the intersection property and clean containers, because their index set consists of only one element. Moreover, hieromorphisms are hierarchically quasiconvex, uniformly coarsely lipschitz, and full. Let us prove that comparison maps are not uniform quasi isometries. First notice that, as each hierarchically hyperbolic space has an index set of cardinality one, there is only equivalence class that spans the whole tree T. Let vand ube two vertices in T, at distance d. Then, the comparison map cvÑu:xay Ñ xayis a p|k|d,0q-lipschitz map. Therefore, as |k|ą1 and we cannot bound the distance dbetween two vertices in the unbounded tree T, comparison maps cannot be uniform quasi isometries, as claimed. The above remark shows that Theorem 3.0.1 cannot be applied to show that non-euclidean Baumslag Solitar groups are hierarchically hyperbolic. The following result is an analog of Lemma 1.2.6, it shows that hierarchically hyperbolic groups cannot contain infinite distorted cyclic subgroups. Remark 3.3.9. If Gis a hierarchically hyperbolic group, then Gcannot have a subgroup isomorphic to BSpn, mq “ xa, t |tant´1“amy, with |n| ‰ |m|. Indeed, suppose there is an embedding ι:BSpn, mqãÑG. We have that ιpaqis an infinite order element of G. By [35, Theorem 7.1] and [36, Theorem 3.1], ιpaqis undistorted, which is a contradiction. More generally, if a group Ghas a hierarchical hyperbolic structure, then it cannot be unbalanced, as it cannot contain infinite distorted cyclic subgroups. After examining the above remark, one would be tempted to think that the only way that a Baumslag Solitar group BSpm, nqhas a hierarchically hyperbolic structure precisely when |m| “ |n|. This is indeed, the case, and we devote the last chapter of this thesis to study hierarchical hyperbolicity for a much broader class of groups that we choose to call hyperbolic-2-decomposable groups. Chapter 4 Hierarchical hyperbolicity of hyperbolic-2-decomposable groups In this chapter we will consider groups that split as graphs of groups with 2-ended edge groups. Recall that, if Pis a property of a group, we say that a group is P-2-decomposable if it splits as a graph of groups with 2-ended edge groups and vertex groups satisfying property P. We now state the main result of the chapter. Theorem 4.0.1. Let Gbe a hyperbolic-2-decomposable group. The following are equivalent. 1. Gadmits a hierarchically hyperbolic group structure. 2. Gdoes not contain a distorted infinite cyclic subgroup. 3. Gdoes not contain a non-Euclidean almost Baumslag–Solitar group. Moreover, if Gis virtually torsion-free, condition (3) can be replaced by 3’. Gdoes not contain a non-Euclidean Baumslag–Solitar group. Before we begin with the chapter, we state a few questions and possible future directions. 4.0.1 Questions The non virtually torsion-free case: our results are stated differently for the case of virtually torsion-free groups. The main problem being that we could not determine in the class of hyperbolic-2-decomposable groups whether all non-Euclidean almost Baumslag–Solitar groups contain a Baumslag–Solitar subgroup. Question 4.0.2. Does every non-Euclidean almost Baumslag–Solitar subgroup of a hyperbolic-2decomposable group contain a non-Euclidean Baumslag–Solitar subgroup? 91 92 CHAPTER 4. HYPERBOLIC-2-DECOMPOSABLE GROUPS THAT ARE HHG We stress that this question has a positive answer for certain torsion-free groups. In [59, Proposition 7.5] the author shows that the question has a positive answer for GBS groups. In [28, Proposition 9.6] the author extends the result to (torsion-free hyperbolic)-2-decomposable groups. However, the results appearing in those papers rely heavily on the absence of torsion. As we will see in Section 4.1, it is enough to assume that Gis virtually torsion-free. Moreover, recall that a graph of virtually torsion-free groups may not have a virtually torsion-free fundamental group (Example 1.9.16). Generalization to HHG-2-decomposable In our proofs, hyperbolicity of the edge groups is used only in Theorem 4.2.2 and Lemma 4.2.6. Thus we expect that finding appropriate replacements for the two results above will yield a sufficient condition for a (hierarchically hyperbolic)-2decomposable group to be hierarchically hyperbolic. However, the question becomes harder when asking for a full characterization. As remarked before, all hierarchically hyperbolic groups are balanced, hence balancedness is surely a necessary condition in Question 2. Question 4.0.3. Under which conditions a (hierarchically hyperbolic)-2-decomposable group is hierarchically hyperbolic? A possible strategy to answer this question would be to extend the tools developed in Section 4.2 to the class of hierarchically hyperbolic groups. That is to say, provide conditions guaranteeing that the hierarchically hyperbolic structure of edge groups can be included in the one of the vertex group. However, we don’t think this strategy would work in the general case. For instance, consider Z22-decomposable groups (also known as tubular groups). If one vertex has three incoming edges, defining pairwise linearly independent lines, there is no straightforward way of defining a hierarchically hyperbolic group structure on Z2that contains each edge group. 4.0.2 Balanced groups A fundamental notion throughout the chapter is the notion of balanced group. Definition 4.0.4. Let Gbe a group and gPG. We say that gis balanced either if ghas finite order, or if whenever gnis conjugate to gm, it must follow |n|“|m|. We say that a group Gis balanced if every element is balanced. Lemma 4.0.5 ([90, Lemma 4.14]).Let Gbe a group and assume that there exists a balanced subgroup Hof Gof finite index. Then, Gis balanced. We are now going to study how balanced groups behave under amalgamated products and HNN extension over virtually cyclic groups. A key property of virtually cyclic groups that will be used throughout the chapter is that if a, b are infinite order elements of a virtually cyclic group, then there are N, M such that aN“bM. Lemma 4.0.6. Let Cbe a virtually cyclic group and G“A˚CB. Then Gis balanced if and only if A, B are. 93 Proof. One implication is clear. To show the converse, let gPGbe an infinite order element and let hPGbe such that hgnh´1“gmfor |n|‰|m|. If gis acts hyperbolically on the Bass-Serre tree Tcorresponding to G, then the translation length `Gpgqis positive. Moreover, `Gpgnq“|n|`Gpgq and `Gphgh´1q “ `Gpgq. Thus, if hgnh´1“gmthen |n|“|m|, which is a contradiction. Thus, we can assume that gacts elliptically on T. Therefore, there exists xsuch that xgx´1belongs in Aor B. Assume without loss of generality that xgx´1PA. We have (4.1) pxhx´1qpxgx´1qnpxhx´1q´1“ pxgx´1qm. If we write a“ pxgx´1q P Aand k“xhx´1, Equation (4.1) becomes kank´1“am. Write kin normal form k0¨¨¨ks, where kiPA´1 or B´1. We have pk0¨¨¨ksqaT npk0¨¨¨ksq´1a´T m “1. There are now two cases. First, assume that no powers of acan be conjugated into C, for instance, this happens whenever |C|ď8. Then by the normal form theorem, s“0k0PAand hence A was not balanced. So suppose that there is some power aof athat can be conjugated into C. Up to conjugating a and kand taking powers of a, we can assume that aPCand kank´1“amholds. Again, consider the normal form k“k0. . . ks. We will proceed by induction on s. Case s“0. In this case we have k0ank´1 0“am. Since aPC, if k0PA(resp. B), we have that A (resp. B) is unbalanced. Induction step. Suppose that the claim holds for kwith normal-form length s´1. We will show that it holds for length s. Consider the equation kank´1“amand assume that khas normal-form length s. Observe that for each Tthe equation kaT nk´1“aT m still holds. We will show that, for Tlarge enough, we can write kaT nk´1“aT m as k1cn1pk1q´1“cm1with cPC,|n1| ‰ |m1|and k1 with normal-form length at most s´1. Then we are done by induction hypothesis. We have pk0¨¨¨ksqanpk0¨¨¨ksq´1“am. By the normal form theorem, b“ksank´1 sPC. Since Cis 2-ended, there is cPCand P1, P2, P3, P4 such that aP1“cP2and bP3“cP4. Let K“k0¨¨¨ks´1. Then we have (4.2) KksaP1P3nk´1 sK´1“aP1P3m 94 CHAPTER 4. HYPERBOLIC-2-DECOMPOSABLE GROUPS THAT ARE HHG Let’s focus on the left-hand side only, conjugating it by K. We have kscP2P3nk´1 s“ksaP1P3nk´1 s“bP1P3“cP1P4. Since ksbelongs to either Aor B, all the elements of the above series of equations are in one between A, B, say A. Since Ais balanced, we need to have |P2P3n|“|P1P4|. Thus, up to possibly substituting nwith ´n, we can write the left-hand-side of Equation (4.2) as KcP2P3nK´1. Now, applying the equality aP1“cP2to the right-hand-side of Equation (4.2), we have KcP1P4K´1“KcP2P3nK´1“cP2P3m. We are now done by induction hypothesis. By applying repeatedly the previous lemma, we obtain the following corollary. Corollary 4.0.7. If Gis a balanced-2-decomposable group such that the underlying graph is a tree, then Gis balanced. It is straightforward to check that HNN extensions of balanced groups are not balanced in general: Simply consider BSp2,3qas the HNN extension xa, t |ta2t´1“a3y – xay˚ta2t´1“a3. To finish this subsection we include results that give sufficient conditions for an HNN extension over a balanced group to be balanced. We stress that these results are modified versions of [28, Proposition 6.3] and [28, Theorem 6.4]. They have been modified as to allow torsion. Proposition 4.0.8. Let Hbe a balanced group, A, B ďHbe virtually cyclic subgroups and φ:AÑBbe a isomorphism. Let G“H˚φ. Then. 1. If gPHbut no power of gis conjugate in Hinto AYBthen gis still balanced in G. 2. If Aand Bare non-commensurable in H, then Gis also a balanced group. Proof. Suppose gwas not balanced in G. Hence there is hPG´Hsuch that hgph´1“gqfor some |p| ‰ |q|. Since hPG´H, we can write h“h1tε1. . . hr´1tεrhrin reduced form. By assumption hrgh´1 rdoes not belong to Anor B, and hence hgqh´1cannot represent an element of H. Thus, hPHand since His balanced |q|“|p|. For the second item, we only need to check the balancedeness of elliptic elements in G, since a translation length argument similar to that of Lemma 4.0.6 rules out unbalancedeness of hyperbolic elements. Thus, if Gis unbalanced, by the first item there must exist an unbalanced infinite order element hPHsuch that some power of hcan be conjugated into AYB. Therefore, we can assume without loss of generality that hPAYB. Assume that h“aPA. Since ais unbalanced, there is some gPGsuch that gaig´1“ajwith |i|‰|j|. Let g“h1tε1. . . hrtεrbe the reduced form 95 expression in G. Since ghig´1“hjhas normal form length 1, there must exist some possible reduction in ph1tε1. . . hrtεrqhiph1tε1. . . hrtεrq´1. There are two possible ways that this could happen: Either εr“1 and hrhih´1 rPAor εr“ ´1 and hrhih´1 rPB. If the latter occurs, then the proof is complete, as hrhih´1 ris an infinite order element in AhrXB. Assume now that the former case occurs. Since Ais a 2-ended balanced group, there must exist ksuch that hraikh´1 r“a˘ik. Therefore, tεrhraikh´1 rt´εr“ta˘ikt´1“b˘ik. Again, as before, we have two possibilities: Either hr´1b˘ikh´1 r´1belongs in Band εr´1“ ´1 or hr´1b˘ikh´1 r´1belongs in Aand εr´1“1. If the latter occurs, the proof is complete. If the former occurs, since Bis a 2-ended balanced group, then hr´1b˘ikk1h´1 r´1“b˘ikk1for some k1. We can continue performing reductions in the expression of gaig´1and at each step we have the same dichotomy where either the proof is complete or we can continue reducing. Note that at some point of the reduction we obtain hisuch that AhiXBor AXBhiis infinite. Indeed, otherwise for some K‰0 the equality gaKig´1“aKj would hold for |Ki|“|Kj|, contradicting the assumption. Corollary 4.0.9. Let Gbe an HNN extension of the balanced group Hwith stable letter tand 2-ended associated subgroups Aand Bof H. Let aPA, b PBbe infinite order elements such that tat´1“b. Moreover, suppose that there is hPHconjugating a power of ato a power of b, so that haih´1“bj. Then Gis balanced if and only for every pair of elements a, b as above we have |i|“|j|. Proof. One implication is clear, we now show that Gis balanced provided that for every hPH such that haih´1“bjfor some i, j it follows that |i|“|j|. Assume that Gis an unbalanced group. Therefore, by the second assertion in the previous proposition, there must exist some h1PHsuch that AXh1Bh1´1is infinite. Since HNN extensions are defined up to conjugation of the corresponding embedding maps, by conjugating by h1we can assume that AXBis infinite in H. By the first assertion in the previous proposition, the only elements that can be unbalanced are those hPHthat can be conjugate in Hinto AYB. Thus, we can assume without loss of generality that the unbalanced elements in Gbelong in AYB. Therefore, if Gis unbalanced, we can assume that for some aPAthere is some gPGsuch that gang´1“amfor some |n| ‰ |m|. We will induct on the length of the reduced form of gto show that gang´1“amimplies |n|“|m|, obtaining a contradiction. Let g“h0tε1h1. . . tεrhrbe the reduced expression of g. Let us say that rdenotes the reduced form length of g. Assume that r“0. That is to say, gPH. Since His balanced, we have |n|“|m|. Assume now that the claim holds for elements of reduced form length r´1, and let gof reduced form length rbe such that gang´1“am. We denote by bPBthe element such that tat´1“b. Note that if the equation gang´1“amholds in G, then for every Twe have that gaT ng´1“aT m for every Tą0. Since the element gang´1“ambelongs in H, by the normal form theorem, 96 CHAPTER 4. HYPERBOLIC-2-DECOMPOSABLE GROUPS THAT ARE HHG gang´1must admit some reduction in its reduced form. There are two ways that this reduction can occur: Either εr“1 and hranh´1 rbelongs in Aor ε1“ ´1 and hranh´1 rbelongs in B. Note that in the former case, since Ais 2-ended and balanced, there must exist some ksuch that hraknh´1 r“a˘kn. Therefore, tε rhraknh´1 rt´εr r“b˘kn. In the latter case we have that hranh´1 r“b1PB. Since Bis a 2-ended group, there must exist l1, l2such that pb1ql1“bl2. Thus, hranl1h´1 r“ pb1ql1“bl2. By assumption, we must have that |nl1| “ |l2|. Therefore, in the latter case we have that t´1hranl1h´1 rt“t´1b˘l2t“a˘l2“a˘nl1. In both cases, we use the induction step to conclude |kn| “ |km|or |l1n| “ |l1m|respectively. In particular, since k‰0‰l1, we conclude |n|“|m|. 4.0.3 Convexity In this chapter, we will make use of two notions of convexity. The first one, called hierarchical quasiconvexity, heavily relies on the hierarchical structure. For instance, it is not quasi-isometric invariant. For a more precise account, we refer to [73]. To detect hierarchical quasiconvexity sometimes it is convenient to check a stronger property. Definition 4.0.10 (Strong quasiconvexity). A subset Yof a quasigeodesic space Xis said to be strongly quasiconvex if there is a function M:r1,8q Ñ Rsuch that every λ–quasigeodesic in Xwith endpoints in Ystays Mpλq–close to Y. Theorem 4.0.11 ([73, Theorem 6.3]).Let pG, Sqbe a hierarchically hyperbolic group and YĎG be a subset. Then if Yis strongly quasiconvex, it is hierarchically quasiconvex, where the constants determine each other. A special case of strongly quasiconvex subsets is given by peripheral subgroups of relatively hyperbolic groups. Lemma 4.0.12 ([34, Lemma 4.15]).Let Pbe a peripheral subgroup in the relatively hyperbolic group G. Then Pis strongly quasiconvex. In the case of hyperbolic spaces, relative hyperbolicity and strong quasi-convexity are intimately related. Definition 4.0.13. We say that a collection of subgroups tHiun i“1of Gis almost-malnormal if HiXgHjg´1is finite unless i“jand gPHi. Theorem 4.0.14 ([22, Theorem 7.11]).Let Gbe a hyperbolic group and tHiun i“1be a finite family of subgroups of G. Then Gis hyperbolic relative to tHiuif and only if tHiuis an almost-malnormal family of strongly quasiconvex subgroups. Definition 4.0.15 (Glueing hieromorphism). Let pH, S1qand pG, S2qbe hierarchically hyperbolic groups. A glueing hieromorphism between Hand Gis a group homomorphism φ:HÑG 4.1. HIERARCHICAL HYPERBOLICITY OF (2-ENDED)-2-DECOMPOSABLE GROUPS 97 that can be realized as a full hieromorphism pφ, φ♦, φ˚ Uqsuch that the image φpHqis hierarchically quasi-convex in Gand the maps φ˚ U:CUÑCφ♦Uare isometries for each UPS1. If the map φ:HÑGis injective, we say that the glueing hieromorphism is injective. 4.1 Hierarchical hyperbolicity of (2-ended)-2-decomposable groups In this section, we focus on (2-ended)-2-decomposable groups. That is to say, graphs of groups where every vertex and edge group is 2-ended. We begin the section by recalling some useful results on 2-ended groups. 4.1.1 Two-ended groups In this subsection, we recall basic results and remarks on the structure of two-ended groups. An important result of these type of groups is known as the structure theorem for infinite virtually cyclic groups. Throughout the chapter, we will make use of this fact on many occasions. Lemma 4.1.1 ([89, Lemma 4.1]).If Gis an infinite virtually cyclic group, then either 1. Gadmits a surjection with finite kernel onto the infinite cyclic group Z, or 2. Gadmits a surjection with finite kernel onto the infinite dihedral group D8 We recall that the infinite dihedral group is the group defined by the presentation D8“ xr, s | srs “r´1, s2y. Note that every element of D8can be written as srk, for P t0,1uand kPZ. Moreover, every element of the form srkhas order 2, and an element of the form rkhas infinite order precisely when k‰0. Using those observations, we have the following Lemma. Lemma 4.1.2. Let Gbe a virtually cyclic group. Let Φ1,Φ2:GÑD8be homomorphisms with finite kernel and finite index image. Then KerpΦ1q “ KerpΦ2q. Proof. As before, D8“ xa, b |bab “a´1, b2y. Suppose that there is gPGsuch that gPKerpΦ1q and gRKerpΦ2q. Since gPKerpΦ1q, we conclude that ghas finite order, otherwise |KerpΦ1q|“ 8. Since Φ2pGqhas finite index in D8there exists cPGsuch that Φ2pcqhas infinite order. In particular there exist k1PZ, k2PZ´ t0usuch that Φ2pgq “ bak1and Φ2pcq “ ak2, and so Φ2pgcq “ bak1`k2. Again, gc has to have finite order to not contradict |KerpΦ2q|ă 8 . However, since gPKerpΦ1qwe have that Φ1pgcq “ Φ1pcq, and so gc cannot have finite order. From this we conclude KerpΦ1q Ď KerpΦ2q. The symmetric argument yields the claim. Remark 4.1.3. Note that an infinite virtually cyclic group Gcannot surject onto both Zand D8 with finite kernel. Indeed, assume that two surjective homomorphisms Φ : GÑZand Φ1:GÑD8 104 CHAPTER 4. HYPERBOLIC-2-DECOMPOSABLE GROUPS THAT ARE HHG 2. if π1pGqis virtually torsion-free then π1pGqmust contain a non-Euclidean Baumslag-Solitar subgroup. Proof. By definition of balanced edges (Definition 4.1.15), if eis unbalanced and φ˘are the monomorphisms associated to the edge e, then there exists an infinite order element a1PGe and hPπ1pG´eqsuch that hφ`pa1qih´1“φ´pa1qjfor some |i|‰|j|. Let adenote φ`pa1q and sdenote tehfor short. By assumption, ahas infinite order, and so s‰1. Then xa, syis a non-Euclidean almost Baumslag-Solitar group. If, in addition, π1pGqis virtually torsion-free then there exists Ną1 such that aNand sNbelongs in a torsion-free subgroup of π1pGq. Note that sNaN¨iNs´N“sN´1pspaiqN¨iN´1s´1qs´pN´1q“ “sN´1ppajqN¨iN´1qs´pN´1q“ “sN´2pspaiqJN¨iN´2s´1qs´pN´2q“ “ ¨¨¨ “ aN¨jN Therefore, the relation sNpaNiNqs´N“aNjNis satisfied in a torsion-free subgroup Qof π1pGq. By Lemma 4.1.5, Qis a generalized Baumslag-Solitar group. Since NiN{NjN“ pi{jqN‰ ˘1, by [59, Proposition 7.5] the subgroup xaN, sNycontains some non-Euclidean Baumslag-Solitar group. Combining Lemma 4.1.19 with Corollary 4.1.22 we obtain Theorem 1.9.15 from the introduction: Theorem 4.1.23. Let Gbe a graph of groups where none of the vertex groups contain distorted cyclic subgroups. Then π1pGqcontains a non-Euclidean almost Baumslag-Solitar subgroups if and only if Ghas an unbalanced edge. Proof. If G“π1pGqcontains a non-Euclidean almost Baumslag-Solitar subgroup then it is unbalanced. By Lemma 4.1.19 we obtain that Gmust contain some unbalanced edge. Corollary 4.1.22 shows the converse. We are now ready to prove the main result of this section. Theorem 4.1.24. Let Gbe a graph of groups, where all vertex and edge groups are two-ended. Assume moreover that π1pGqis virtually torsion-free. Then the following are equivalent. 1. π1pGqadmits a hierarchically hyperbolic groups structure. 2. Gis linearly parametrizable. 3. π1pGqis balanced. 4.1. HIERARCHICAL HYPERBOLICITY OF (2-ENDED)-2-DECOMPOSABLE GROUPS 105 4. π1pGqdoes not contain BSpm, nqwith |m|‰|n|. 5. π1pGqdoes not contain a distorted infinite cyclic subgroup. Proof. 3ô2 By Corollary 4.1.18 we have that π1pGqis linearly parametrizable if and only if every edge ein Gis balanced. Moreover, by Lemma 4.1.19 we have that every edge in Gis balanced if and only if π1pGqis balanced. 5ñ3 Assume that π1pGqis unbalanced. Therefore, by Lemma 4.1.19 there is an edge e, an infinite order element aPGeand an element hPπ1pG´eqsuch that hφ`paqih´1“φ´paqj, with |i| ‰ |j|. Let x“φ`paqand y“φ´paq. Since eis unbalanced, there is a spanning tree that does not contain e. In particular, we can assume there is a stable letter tassociated to the edge e such that tyt´1“x. We claim that xxyis distorted. Note that xis of infinite order. To simply notation, we will write A«rBif |A´B| ď r. We have: d`1, xN¨i˘«2|h|d`1, hxN¨ih´1˘“d`1, yN¨j˘«2|t|d`1, xN¨j˘. This is to say, for each Nwe have ˇˇd`1, xN¨i˘´d`1, xN¨j˘ˇˇď2p|h|`|t|q. Since |i|‰|j|, it is now a standard argument to show that xxyis distorted. Indeed, restating the argument before for a general exponent Mwe have d´xM, xt|j| |i|Mu¯ď |h|`|t|`i. Assuming that |i|ą|j|, we can iterate the inequality above to obtain that dp1, XMqis comparable to log|j| |i|pMq¨p|h|`|t| ` iq. That is to say, dp1, XMqgrows logarithmically, showing that the map nÞÑ xncannot be a quasi-isometric embedding. 4ñ3 Assume that π1pGqis unbalanced. Therefore, by Lemma 4.1.19, Gmust contain an unbalanced edge. The second item of Corollary 4.1.22 concludes the proof. 1ñ5 Follows from [35, Theorem 7.1] and [36, Theorem 3.1]. 2ñ1 Follows from Theorem 4.1.12. 5ñ4 Since non-Euclidean Baumslag-Solitar groups contains distorted cyclic subgroups if Gcontains some non-Euclidean Baumslag-Solitar subgroup we obtain the result. Theorem 4.1.25. Let Gbe a graph of groups, where all vertex and edge groups are two-ended. Then the following are equivalent. 1. π1pGqadmits a hierarchically hyperbolic groups structure. 106 CHAPTER 4. HYPERBOLIC-2-DECOMPOSABLE GROUPS THAT ARE HHG 2. Gis linearly parametrized. 3. π1pGqis balanced. 4. π1pGqdoes not contain a non-Euclidean almost Baumslag-Solitar subgroup. 5. π1pGqdoes not contain a distorted infinite cyclic subgroup. Proof. Assume that π1pGqis unbalanced. Therefore, by Lemma 4.1.19, Gmust contain an unbalanced edge. The first item of Corollary 4.1.22 shows the implication 4 ñ3. The rest of the implications are the same as in Theorem 4.1.24. 4.2 Hierarchical hyperbolicity of hyperbolic-2-decomposable groups In this section, we give a necessary and sufficient condition for the fundamental group of a graph of groups with hyperbolic vertex groups and virtually cyclic edge groups to be a hierarchically hyperbolic group. We do so by extending the tools introduced in the previous section. To that end, we make use of Theorem 4.2.2 to induce a hierarchically hyperbolic group structure on the groups Gv. We begin by showing the following lemma. This allows us, without loss of generality, to restrict our attention to graphs of hyperbolic groups with infinite virtually edge groups. Lemma 4.2.1 (Dealing with finite vertices/edges). Let Gbe a graph of groups such that π1pGqis infinite and Ghas hyperbolic vertex groups and virtually cyclic edge groups. Then there exists a finite graph of groups G1with infinite hyperbolic vertex groups and 2-ended edge groups such that π1pG1q “ π1pGq. Proof. Given a graph of groups Hlet FpHqbe the set of edges with finite associated edge group, that is tePEpHq||Ge| ď 8u. Let G0“G. We will produce a sequence of graph of groups Gisuch that π1pGiq – π1pGq,Gihas hyperbolic vertex groups and virtually cyclic edge groups and |FpGiq| ă |FpGi´1q|. Since the graph of groups is finite, eventually we will find Gnsuch that FpGnq“H. In particular, if Gnhas at least one edge, then the associated edge group is infinite. Hence, the vertex groups needs to be infinite and we are done. If there are no edges, then there is a single vertex labelled by π1pGq, which is hyperbolic by construction. Since, by assumption π1pGq is infinite, we are done. Suppose Giis defined. Firstly, suppose that there is ePFpGiqsuch that there exists a spanning tree Teof Gicontaining e(recall that π1pGqdoes not depend on the choice of spanning tree, as pointed out in Remark 1.3.4). Then the subgroup Ge`˚GeGe´is hyperbolic by Theorem [17, Corollary Section 7]. Then let Gi`1be defined from Giby replacing the edge eand the incident vertices by 4.2. HIERARCHICAL HYPERBOLICITY OF HYPERBOLIC-2-DECOMPOSABLE GROUPS107 a single vertex with associated group Ge`˚GeGe´, and leaving the other edge maps unchanged. By doing this, we still have hyperbolic vertex groups and virtually cyclic edge groups. So, suppose that no element of FpGiqcan be included in a spanning tree. This is to say that all elements of FpGiqare loops. Let ePFpGiq, and let vbe the vertex incident to it. Then by [18, Corollary 2.3], the HNN extesion Gv˚Geis hyperbolic. Then we define Gi`1as the graph of groups obtained from Giby removing the edge eand changing the vertex group of vto Gv˚Ge. From now on, whenever we state a result on a graph of hyperbolic groups Gwe will always assume that the associated edge groups Geare virtually cyclic and infinite. In other words, from now on we assume that the groups considered are hyperbolic-2-decomposable. Given a vertex group Gv, one of the main challenges that we have to face in this setting is the fact that the incoming edge groups do not necessarily form an almost-malnormal collection in Gv (Definition 4.0.13). As a consequence, these edge groups may not be geometrically separated so as to include them in the hierarchical hyperbolic structure of Gv. The following theorem solves this problem, and it is pivotal in the proof of the main theorem in this section. We also stress that it is a consequence of [14, Theorem 9.1]. Theorem 4.2.2. Let Gbe a group hyperbolic relative to a family of hierarchically hyperbolic groups tpHi,Siqun i“1. Suppose that there is a finite family of subgroups tKαuαPΛand homomorphisms φα:KαÑGsuch that for each αthere exists iand gPGsuch that φαpKαqhas finite index in Hg i. Finally, suppose that each group Kαis equipped with a hierarchically hyperbolic structure Kα such that φg´1 α:pKα,KαqÑpHi,Siqis a glueing hieromorphism. Then there is a hierarchically hyperbolic structure pG, Sqon Gsuch that φαis a glueing hieromorphism for every α. Moreover, if all pHi,Siqsatisfy the intersection property, so does pG, Sq, and similarly for clean containers. Proof. This theorem is an adaptation of Theorem 4.2.2. We will follow almost verbatim the part of the proof that describes such a structure on G, but we will not verify the axioms as it will not add clarity to the current proof. We will conclude the proof by showing that the maps φαcan be realized as glueing hieromorphisms. The structure: For each i“1...,n and each left coset of Hiin G, fix a representative gHi. Let gSibe a copy of Siwith its associated hyperbolic spaces and projections in such a way that there is a hieromorphism HiÑgHiequivariant with respect to the conjugation isomorphism HiÑHg i. Let p Gbe the hyperbolic space obtained by coning-off Gwith respect to the peripherals tHiu, and let S“ tp GuYŮgPgŮiSgHi. The relation of nesting, orthogonality or transversality between hyperbolic spaces belonging to the same copy SgHiare the same as in SHi. Further, if U, V belong in two different copies of different cosets, then we impose transversality between them. Finally, for every UPSgHiwe declare that Uis nested into p G. The projections are defined as follows: πp G:GÑp Gis the inclusion, which is coarsely surjective 108 CHAPTER 4. HYPERBOLIC-2-DECOMPOSABLE GROUPS THAT ARE HHG and hence has quasiconvex image. For each UPSgHi, let ggHi:GÑgHibe the closest-point projection onto gHiand let πG U“πHi U˝ggHi, to extend the domain of πUfrom gHito G. Since each πHi Uwas coarsely Lipschitz on CUwith quasiconvex image, and the closest-point projection in Gis uniformly coarsely Lipschitz (Lemma 1.4.6), the projection πG Uis uniformly coarsely Lipschitz and has quasiconvex image. For each U, V PSgHi, the various ρV Uand ρU Vare already defined. If UPSgHiand VPSg1Hj, then ρU V“πVpgg1HjpgHiqq. Finally, for U‰p G, we define ρU p Gto be the cone-point over the unique gHiwith UPSgHi, and ρp G U:p GÑCUis defined as follows: for xPG, let ρp G Upxq “ πG Upxq. If xPp Gis a cone point over g1Hj‰gHi, let ρp G Upxq “ ρSg1Hj U, where Sg1Hjis the Ď–maximal element of Sg1Hj. The cone-point over gHimay be sent anywhere in CU. By [14, Theorem 9.1], the construction above endows pG, Sqwith a hierarchically hyperbolic group structure. Hieromorphisms: Fix α. By assumption there exists iand gPGsuch that φαpKαq Ď Hg i. Moreover, Φα“φg´1 α:pKα,Kαq Ñ pHi,Siqis a glueing hieromorphism. Our goal is to show that φ:pKα,KαqÑpG, Sqcan be equipped with a glueing hieromorpism structure. To simplify notation we will drop the αand isubscript and denote pK, Kq“pKα,Kαq,φ“φα, pH, SHq“pHi,Siqand so on. For every VPK, define φ♦pVq “ gΦ♦pVqand φ˚ V“g˚˝Φ˚ V, where g˚is the isometry associated to the multiplication gPG. By assumption, the maps Φ˚ V:CVÑCΦ♦Vare isometries, and for each UPSH, the space CHUand the space CGgU are isometric. Thus, the maps φ˚ Vare isometries. We need to show that the following two diagrams coarsely commute. Kφ// πK V  G πG φ♦pVq  CVφ˚ U //Cφ♦pVq CVφ˚ V// ρV U  Cφ♦pVq ρφ♦pVq φ♦pUq  CUφ˚ U //Cφ♦pUq This is a matter of unwinding the definitions. We will check the first one, the second is analogous. So, let xPK. Recall that φpxq “ gΦpxqg´1PgHig´1. Then πG φ♦pVqpφpxqq “ g˚˝πHi Φ♦pVq˝g´1˝ “ g˚˝πHi Φ♦pVqpggHipΦpxqg´1qq. (4.5) Note that dpΦpxqg´1, gHiq ď |g|. Since all the map are coarsely Lipschitz, there is a uniform bound between πHi Φ♦pVqpggHipΦpxqg´1qq and πHi Φ♦pVqpΦpxqq. That is, up to a uniformly bounded error, we can write Equation 4.5 as (4.6) πG φ♦pVqpφpxqq “ g˚´πHi Φ♦pVqpΦpxqq¯. 4.2. HIERARCHICAL HYPERBOLICITY OF HYPERBOLIC-2-DECOMPOSABLE GROUPS109 On the other hand, we have (4.7) φ˚ V˝πK Vpxq “ g˚`Φ˚ U˝πK Vpxq˘. Since g˚is an isometry, Equations (4.6) and (4.7) give the result. Note that the constant of the coarse commutativity depend on g. However, since there are only finitely many pairs pKα, Hiq, we obtain uniformity. Hence, the map φcan be equipped with a hieromorphism structure. By construction, the maps φ˚ Uare isometries, and the hieromorphism is full. To see that it has hierarchically quasiconvex image, observe that its image is at finite Hausdorff distance from a peripheral subgroup, hence it is strongly quasiconvex (Lemma 4.0.12). Then it is hierarchically quasiconvex by Theorem 4.0.11. [73, Thorem 6.3]. Intersection property and clean containers: We start by checking clean containers, that is to check that for each UĎTPSwe have UKcontT KU. If U“p Gthere is nothing to check. Hence, assume UPgSiand let gSibe the Ď–maximal element of gSi. Recall that the relations on Sare defined such that if U, V PS´tp Guare not transverse, then there is iP t1, . . . , nuand gPGsuch that U, V PgSi. In particular, UKVimplies U, V PgSi. Hence, cont p G KU“contgSi KU. Moreover, if UĎTand T‰p G, it follows TPgSi. Since we assumed that pHi,Siqhas clean containers, we have UKcontT KUfor all TPgSi, completing the proof. Consider now the intersection property. By hypothesis, for each gSithe map ^gHiis defined. Then define ^:pSYtHuqˆpSYtHuq Ñ pSYtHuq by considering the symmetric closure of the following: U^V“$ ’ ’ ’ ’ & ’ ’ ’ ’ % Uif V“p G U^gHiVif U, V PgSifor some i, g Hotherwise. The only property to verify that does not follow directly is to check that if UPgSiand VPg1Sj with gSi‰g1Sj, then there is no Wnested in both U, V . But if such a Wexisted, then it needs to belong to both gSiand g1Sj, a contradiction. 4.2.1 Commensurability and conjugacy graph In this subsection we extend the results obtained in Section 4.1 to the general setting. The key object that will allow us to do this is the conjugacy graph (Definition 4.2.10). This is a graph of groups that, combined with Theorem 4.2.2, provides vertex groups with a hierarchical hyperbolic structure realizing edge maps as glueing hieromorphisms. As the vertex groups in the graphs of groups considered are not 2-ended, the whole graph of groups cannot be linearly parametrized. Moreover, the edge groups do not necessarily embed into vertex groups in an almost malnormal way. To overcome those problems, we will consider the elementary 110 CHAPTER 4. HYPERBOLIC-2-DECOMPOSABLE GROUPS THAT ARE HHG closure of subgroups. A systematic study of elementary closures of WPD subgroups (which include cyclic subgroups of hyperbolic groups as a special case) is carried on in [30], where the authors show such subgroups needs to be hyperbolically embedded in the ambient group. For the sake of self-containment, we recall some useful properties of the elementary closure. Definition 4.2.3 (Elementary closure). Let Gbe a group and let Hbe a subgroup of G. We define the elementary closure of Hin Gas the subgroup EGpHq“tgPG|dHauspgH, Hq ă 8u. Lemma 4.2.4. Let H, K be subgroups of Gsuch that HXKhas finite index in both Hand K, then KďEGpHq. Proof. Let kPKand hPH. Our goal is to uniformly bound dpkh, Hq. Since HXKhas finite index in H, there is k0PHXKat uniformly bounded distance from h. Note that kk0PK. Since HXKhas finite index in K, there is h0PHXKat uniformly bounded distance from kk0. By triangular inequality, we get a uniform bound on dpkh, h0q. Recall that two groups H, K are said to be commensurable if HXKis of finite index in both H and K. In this chapter we adopt a different, more broad notion of commensurability. Definition 4.2.5. Let Gbe a group and A, B ďGbe subgroups. We say that Aand Bare commensurable if there exists gPGsuch that gAg´1XBhas finite index in Band AXg´1Bg has finite index in A. Moreover, we say that two elements a, b PGare non-commensurable if xayand xbyare noncommensurable in G. Note that, in general, Hwill not have finite index in EGpHq. A simple example of this is given by considering the subgroup xayin xay‘xby – Z2. Indeed, in this case we would have EZ2pxayq “ Z2. This is not the case, however, for 2-ended subgroups of hyperbolic groups. Lemma 4.2.6 ([30, Lemma 6.5]).Let Gbe a hyperbolic group and Hbe a 2-ended subgroup. Then EGpHqis 2-ended. In particular, observe that EGpHqhas to be the maximal cyclic subgroup containing H. This yields the following useful lemma. Lemma 4.2.7. Let H1, . . . , Hnbe 2-ended subgroups of a hyperbolic group G. Then 1. Hiand Hjare commensurable in Gif and only if EGpHiqand EGpHjqare conjugate to each other. 2. tEGpH1q, . . . , EGpHnqu is an almost malnormal collection if and only if Hiand Hjare noncommensurable for every i‰j; 4.2. HIERARCHICAL HYPERBOLICITY OF HYPERBOLIC-2-DECOMPOSABLE GROUPS111 Proof. Since Hihas finite index in EGpHiq, we have that EGpHiqand EGpHjqare commensurable if and only if Hiand Hjare. In particular, this shows one implication. Suppose that EGpHiqand EGpHjqare commensurable. Up to conjugate one of them we have that gEGpHiqg´1XEGpHjq has infinite index in both gEGpHiqg´1, and EGpHjq. By Lemma 4.2.4 we have gEGpHiqg´1ď EGpEGpHjqq “ EGpHjqand, by symmetry, EGpHjq ď gEGpHiqg´1. Hence, EGpHiqand EGpHjq are conjugate. For the second item, observe that if EGpHiqand EGpHjqare not commensurable, since they are 2-ended groups it must follow |EGpHiqXgEGpHjqg´1|ď8for all gPG. Hence they are almost malnormal. We now introduce the conjugacy graph associated to an edge group. Definition 4.2.8 (Commensurability class). Let Gbe a group and let Pbe a collection of 2ended subgroups of G. We denote by «the equivalence relation on Pinduced by commensurability. That is to say, P1«P2whenever P1, P2are commensurable (as in Definition 4.2.5). For each PPP we use JPKto denote its commensurability class. Definition 4.2.9 (Equivalence class). Let Gbe a graph of groups with 2-ended edge groups. Consider the multiset U“ tφe`pGeq, φe´pGeq | ePEpΓqu of all the images of edge groups into vertex groups counted with repetitions. Let „0be the relation on Udefined by imposing H1„0H2whenever either there exists esuch that H1“φe`pGeqand H2“φe´pGeq, or H1, H2PGvfor some vand H1«H2in Gv. Extend „0to an equivalence relation „on Uby taking the transitive closure of „0. For a vertex group H, we denote by rHsits equivalence class with respect to „. Definition 4.2.10 (Conjugacy graph). Let Gbe a graph of groups with 2-ended edge groups and let rHsbe the equivalence class of an edge group in G. We define the conjugacy graph associated to rHsas the graph of groups ∆rHsdefined as follows. For each vertex group GvPG, let rHsv“ tH1P rHs | H1ďGvu. Vertices: For each vertex vof the original graph Gand commensurability class JKKof rHsv, add one vertex vKto ∆rHs. Choose once and for all a representative KPJKKand define EGvpKqto be the vertex group associated to vK. Edges: For each edge ePΓ such that φe`pGeq P rHs, add an edge between Jφe`pGeqKand Jφe´pGeqK, with associated edge group Ge. To define the edge maps, let Kbe the chosen representative of Jφe`pGeqK. Then there is hPGe`such that φe`pGeqhĎEGe`pKq. If φe`:GeÑGe` was the edge map of G, let the attaching map of ∆rHsbe defined as φh e`:GeÑEGe`pKq. Note that, by Remark 4.2.7, this map is well defined. 112 CHAPTER 4. HYPERBOLIC-2-DECOMPOSABLE GROUPS THAT ARE HHG Remark 4.2.11. In this chapter, we consider only graphs of groups with 2-ended edge groups. In particular, by Lemma 4.2.6 the vertex groups of the conjugacy graphs are 2-ended. As the edge groups of the conjugacy graphs are the same as the original edge groups, the conjugacy graphs have 2-ended vertex and edge groups. construction. Example 4.2.12. Let F2“ xa, bybe the free group of rank 2 and consider the group Gto be π1pGq “ F2˚ta3t´1“ba2b´1. By construction, the splitting of Ghas one vertex vwith associated vertex group Gv“F2and one edge ewith associated cyclic edge group Ge. We now construct the conjugacy graph ∆rGesassociated to rGes. Note first that the images of the single edge group are commensurable in the vertex group, as bxa3yb´1Xxba2b´1yis infinite. Thus, there is a single conjugacy class of rGesin F2and, therefore, a single vertex in ∆rHs. The associated vertex group of ∆rHsis bEF2pa2qb´1“bxayb´1. There is also a single edge group in ∆rHswith associated edge group equal to the one in G. The associated attaching maps are φe`and φb e´. The conjugacy graph associated to rGesresults in the group xay˚ta2t´1“a3. In the following two lemmas, we describe how is the linear parametrization in a graph of 2-ended groups extended to the general setting using the conjugacy graph. Lemma 4.2.13. Let G–π1pGqbe a graph of hyperbolic groups with 2-ended edge subgroups and let ebe an edge in the underlying graph of G. If ∆rGesdenotes the conjugacy graph associated to rGes, then eis unbalanced in Gif and only if π1p∆rGesqis unbalanced. Proof. Assume first that Gcontains an unbalanced edge e. Therefore, there exists an infinite order element aPGeand hPπ1pG´eqsuch that hφe`paqih´1“φe´paqjfor some |i|‰|j|. By Lemma 1.3.10 there is a path e1, . . . , ekin the graph of G´ewith Aep1q“Gα, Bepkq“Gβsuch that Bhj ejXAej`1is non-trivial for every j“1, . . . , k ´1 (i.e EGe` jpBejqhj“EGe` jpAej`1q) and elements h0PGαand hiPGbpeiqsatisfying (4.8) ptekhk¨¨¨h1h0qφe`paqiptekhk¨¨¨h1h0q´1“φe´paqj, for some |i|‰|j|. This means that the conjugacy graph ∆rGessplits as π1p∆rGes´eq˚te. Recall that by definition the attaching maps in ∆rGesare defined as conjugates φhe1 e1` in Ge1` of the attaching maps φe1` in G. Therefore, since φe`pgq, φe´pg1qare conjugate in π1pGq, following Equation (4.8) we obtain that φe`pgqi“φe´pgqjin π1p∆rGes´eqwhere |i|‰|j|. Assume now that, π1p∆rGesqis unbalanced. We can apply Lemma 1.3.10 to obtain, (4.9) phktk ek¨¨¨h1t1 e1h0qapphktk ek¨¨¨h1t1 e1h0q´1“aq, for some |p| ‰ |q|. Here, ais of infinite order, the various elements hiand abelong to vertex 4.2. HIERARCHICAL HYPERBOLICITY OF HYPERBOLIC-2-DECOMPOSABLE GROUPS113 groups and at least one iis non zero. Our goal is to modify the above equation to obtain an analogous one that holds in π1pGq. Let H0be the vertex group of ∆rGesthat contains aand let H1 be the other vertex group adjacent to e1in ∆rGes(possibly, H0“H1). Let xPH1be such that pt1 e1h0qappt1 e1h0q´1“xin π1p∆rGesq. By definition of conjugacy graphs, there are vertex groups G0, G1of Gsuch that HiďGi. Since the attaching maps in the conjugacy graph are defined as a conjugates of the attaching maps of G, there exists kiPGisuch that the following holds in π1pGq: pk1t1 e1h0k0qappk1t1 e1h0k0q´1“x Let y1“ pk1t1 e1h0k0q. Proceeding in this way, we find an element yk“yof π1pG´eqsuch that yapy´1“aq with |p|‰|q|, showing that eis unbalanced in G. Lemma 4.2.14. Let Gbe a graph of groups with hyperbolic vertices and 2-ended edge subgroups. Suppose, moreover, that for each edge ethe conjugacy graph ∆rGesis linearly parametrizable. Then π1pGqadmits a hierarchically hyperbolic group structure. Proof. For each vertex vPVpGqlet teiube the set of incoming edges and let EpGe` iqbe the elementary closure of the images of the edge groups in Gv. Choose representatives tEiuof the commensurability classes tJEpGe` iqKu. Note that, by Remark 4.2.7, tEiuforms an almost malnormal collection of subgroups. In particular, Gvis hyperbolic relative to tEiuby Theorem 4.0.14. By assumption, the conjugacy graph ∆rGesassociated to rGesis linearly parametrizable for every e. That is to say, for every edge ethere exists ΦrGes:π1p∆rGesq Ñ Dpeq 8such that ΦrGes|Gx:GxÑDpeq 8 is a quasi-isometry, where Gxis either a vertex or edge group of ∆rGes. We endow the various groups Gxwith the hierarchical hyperbolic structure pGx,tDpeq 8uq as described in Lemma 4.1.8. In particular, this allows to equip with a hierarchically hyperbolic group structure every edge group of Gand every group EiďGvas before. Note that this is well defined. Indeed, suppose that e, f are edges incoming in vand Epφe`pGeqq, Epφf`pGfqq are conjugate. Then e„fand hence Epφe`pGeqq and Epφf`pGfqq are identified in the conjugacy graph. Thus the hierarchically hyperbolic structure of the representative Edoes not depend on choices. Finally, note that since the trivial hierarchically hyperbolic structure on D8satisfy the intersection property and clean containers, so do all the hierarchically hyperbolic structures considered thus far. Note that we are now in the hypotheses of Theorem 4.2.2, allowing us to equip every vertex group with a hierarchically hyperbolic structure pGv,Svqthat turn the edge maps into glueing hieromorphisms pGe,SeqãÑ pGv,Svq. Moreover pGv,Svqsatisfy the intersection property and clean containers. Applying Theorem 3.3.1 we obtain that π1pGqis a hierarchically hyperbolic 120 BIBLIOGRAPHY [58] Gilbert Levitt, On the automorphism group of generalized Baumslag-Solitar groups, Geom. Topol. 11 (2007), 473–515. MR2302496 [59] , Quotients and subgroups of Baumslag-Solitar groups, J. Group Theory 18 (2015), no. 1, 1–43. MR3297728 [60] Roger C. Lyndon and Paul E. Schupp, Combinatorial group theory, Springer-Verlag, Berlin-New York, 1977. Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 89. MR0577064 [61] Joseph Maher and Alessandro Sisto, Random subgroups of acylindrically hyperbolic groups and hyperbolic embeddings, Int. Math. Res. Not. IMRN 13 (2019), 3941–3980. MR4023758 [62] Jason F. Manning, The Bowditch boundary of pG, Hqwhen Gis hyperbolic, ArXiv e-prints (April 2015), available at 1504.03630. [63] Howard A. Masur and Yair N. Minsky, Geometry of the complex of curves. I. Hyperbolicity, Invent. Math. 138 (1999), no. 1, 103–149. MR1714338 [64] , Geometry of the complex of curves. II. Hierarchical structure, Geom. Funct. Anal. 10 (2000), no. 4, 902–974. MR1791145 [65] Yoshifumi Matsuda, Shin-ichi Oguni, and Saeko Yamagata, Blowing up and down compacta with geometrically finite convergence actions of a group, ArXiv e-prints (January 2012), available at 1201.6104. [66] Mahan Mj and Lawrence Reeves, A combination theorem for strong relative hyperbolicity, Geom. Topol. 12 (2008), no. 3, 1777–1798. MR2421140 [67] Denis V. Osin, Weak hyperbolicity and free constructions, Group theory, statistics, and cryptography, 2004, pp. 103–111. MR2105438 [68] , Relatively hyperbolic groups: intrinsic geometry, algebraic properties, and algorithmic problems, Mem. Amer. Math. Soc. 179 (2006), no. 843, vi+100. MR2182268 [69] , Acylindrically hyperbolic groups, Trans. Amer. Math. Soc. 368 (2016), no. 2, 851–888. MR3430352 [70] Eliyahu Rips, Subgroups of small cancellation groups, Bull. London Math. Soc. 14 (1982), no. 1, 45–47. MR642423 [71] Bruno Robbio and Davide Spriano, Hierarchical hyperbolicity of hyperbolic-2-decomposable groups, arXiv preprint arXiv:2007.13383 (2020). [72] Derek J. S. Robinson, Recent results on generalized Baumslag-Solitar groups, Note Mat. 30 (2010), no. suppl. 1, 37–53. MR2943040 [73] Jacob Russell, Davide Spriano, and Hung Cong Tran, Convexity in hierarchically hyperbolic spaces, arXiv preprint arXiv:1809.09303 (2018). [74] , The local-to-global property for morse quasi-geodesics, arXiv preprint arXiv:1908.11292 (2019). [75] Jacob Russell and Kate M. Vokes, The (non)-relative hyperbolicity of the separating curve graph, arXiv preprint arXiv:1910.01051 (2019). [76] Michah Sageev, CATp0qcube complexes and groups, Geometric group theory, 2014, pp. 7–54. MR3329724 [77] Jean-Pierre Serre, Trees, Springer-Verlag, Berlin-New York, 1980. Translated from the French by John Stillwell. MR607504 [78] Sam Shepherd and Daniel J. Woodhouse, Quasi-isometric rigidity for graphs of virtually free groups with two-ended edge groups, arXiv preprint arXiv:2007.10034 (2020). [79] Alessandro Sisto, On metric relative hyperbolicity, ArXiv e-prints (October 2012), available at 1210.8081. BIBLIOGRAPHY 121 [80] , Projections and relative hyperbolicity, Enseign. Math. (2) 59 (2013), no. 1-2, 165–181. MR3113603 [81] , Contracting elements and random walks, J. Reine Angew. Math. 742 (2018), 79–114. MR3849623 [82] , What is a hierarchically hyperbolic space?, Beyond hyperbolicity, 2019, pp. 117–148. MR3966608 [83] Davide Spriano, Hyperbolic HHS I: Factor systems and quasi-convex subgroups, ArXiv e-prints (November 2017), available at 1711.10931. [84] , Hyperbolic HHS II: Graphs of hierarchically hyperbolic groups, arXiv preprint arXiv:1801.01850 (2018). [85] John R. Stallings, On torsion-free groups with infinitely many ends, Ann. of Math. (2) 88 (1968), 312–334. MR228573 [86] , On torsion-free groups with infinitely many ends, Ann. of Math. (2) 88 (1968), 312–334. MR228573 [87] Hung Cong Tran, Relations between various boundaries of relatively hyperbolic groups, Internat. J. Algebra Comput. 23 (2013), no. 7, 1551–1572. MR3143594 [88] Kate M. Vokes, Hierarchical hyperbolicity of graphs of multicurves, arXiv preprint arXiv:1711.03080 (2017). [89] Charles T. C. Wall, Poincar´e complexes. I, Ann. of Math. (2) 86 (1967), 213–245. MR217791 [90] Daniel T. Wise, Subgroup separability of graphs of free groups with cyclic edge groups, Q. J. Math. 51 (2000), no. 1, 107–129. MR1760573 [91] , From riches to raags: 3-manifolds, right-angled Artin groups, and cubical geometry, CBMS Regional Conference Series in Mathematics, vol. 117, Published for the Conference Board of the Mathematical Sciences, Washington, DC; by the American Mathematical Society, Providence, RI, 2012. MR2986461 122 BIBLIOGRAPHY Agradecimientos Esta secci´on est´a dedicada a todos aquellos que tuvieron un impacto directo o indirecto en esta tesis. En primer lugar, agradezco a Montserrat Casals-Ruiz, Ilya Kazachkov y Mark Hagen por la orientaci´on y ayuda que me han brindado durante todo el curso de mi formaci´on de doctorado; por la generosidad con la que me han dedicado su tiempo y el invaluable apoyo moral; su energ´ıa y dedicaci´on han sido una inspiraci´on para m´ı. Esta tesis no se hubiera completado sin ellos. Gracias a Jason Behrstock, Ruth Charney, Gustavo Fern´andez-Alcober, Jon Gonz´alez, Alessandro Sisto y Genevieve Walsh por tomarse el tiempo de leer mi tesis y ser parte del jurado. Gracias a Matthew Durham y Alexandre Martin por tomarse el tiempo de leer y evaluar mi tesis. Un agradecimiento adicional para Jason por hospedar mi estad´ıa en CUNY entre septiembre y diciembre de 2019. Gracias a mis coautores, que son excelentes matem´aticos y de quienes he aprendido y sigo aprendiendo. Gracias a FCEyN, UBA por sentar mis bases. Por ofrecer educaci´on p´ublica, de calidad y gratuita a todos y por hacerme sentir en casa siempre que regreso. Gracias a Natalia, que me ayuda a atravesear los momentos dif´ıciles y es mi cable a tierra en los buenos; por todo el amor y por elegirnos cada da. Nunca hubiera estado donde estoy sin el apoyo incondicional y el amor de mi familia. Esta tesis tambi´en est´a dedicada a ellos: a mi mam´a, a mi pap´a, a mis hermanos y padres extendidos, Marcelo y Monica. Gracias a Andr´es, Fidel y Mariano por ser mis amigos desde chicos; por futuras reuniones en VV despu´es de la plaga. Gracias al incre´ıble grupo de humanos del departamento de matem´aticas de la UPV: Albert, Andoni, Elena, Federico, Iker, Marialaura, Matteo, Oihana, Sheila y Xuban. Gracias por el caf´e, los rompecabezas, los juegos de mesa y por acortar los d´ıas largos. Un agradecimiento adicional a la gente de BCAM: Dani, Luz y Javi ˆ2. Finalmente, gracias a Euskadi por ser un gran lugar. Acknowledgements This section is dedicated to all of those who had a direct or indirect impact on this thesis. 123 124 BIBLIOGRAPHY First and foremost, I thank Montserrat Casals-Ruiz, Ilya Kazachkov and Mark Hagen for the guidance and help that they have provided me throughout the course my PhD training; for the generous way they have lent their time to me and the additional moral and truly invaluable support; their energy and dedication has been an inspiration for me. This thesis would not have been accomplished without them. Thanks to Jason Behrstock, Ruth Charney, Gustavo Fern´andez-Alcober, Jon Gonz´alez, Alessandro Sisto and Genevieve Walsh for taking the time to read my thesis and be part of the jury. Thanks to Matthew Durham and Alexandre Martin for taking the time to read and evaluate my thesis. Additional thanks goes to Jason for hosting my stay at CUNY between September and December 2019. Thanks to my coauthors, who are remarkable mathematicians and from whom I have learned and continue to learn. Thanks to FCEyN, UBA for setting my foundations. For offering public, quality, free education to all and for making me feel at home whenever I return. Thanks to Natalia, who manages to keep me going through tough times, and reality checks me in good ones; for the all the love, and for choosing us every day. I would have never be where I am without the unconditional support and love of my family. This thesis is also dedicated to them: to my mom, my dad, my brothers and extended parents, Marcelo and Monica. Thanks to Andr´es, Fidel and Mariano for being my friends since we were little; here’s to hoping for future VV reunions after the plague has dissipated. Thanks to the amazing group of humans in the mathematics department of the UPV: Albert, Andoni, Elena, Federico, Iker, Marialaura, Matteo, Oihana, Sheila, and Xuban. Thanks for all the coffee, the puzzles, the boardgames and for making long days shorter. Additional thanks to the BCAM people: Dani, Luz, and Javiˆ2. Finally, thanks to Euskadi for being an awesome place.