Homeomorphism groups of the cube and other n-manifolds

GROUPS OF PL HOMEOMORPHISMS OF CUBES DANNY CALEGARI AND DALE ROLFSEN D´edi´e `a Michel Boileau sur son soixanti`eme anniversaire. Sant´e! R´ esum´ e. Nous ´etudions les propri´et´es alg´e briques de groupes de PL ou hom´eomorphismes lisses de cubes unitaires dans toutes les dimensions, point par point fixe sur la fronti`ere, et des groupes plus g´en´eralement PL ou lisses agissant sur les vari´et´es et qui fixe ponctuellement une sous-vari´et´e de codimension 1 (resp. codimension 2), et montrent que ces groupes sont localement indicable (resp. de ordonnable circulaire). Nous donnons ´egalement de nombreux exemples de groupes int´eressants qui peuvent agir, et de discuter de certaines autres contraintes alg´ebriques que ces groupes doivent satisfaire, y compris le fait que un groupe d’hom´eomorphismes PL de la n-cube (ponctuelle fixe sur la fronti`ere) ne contient pas ´el´ements qui sont plus que d´eform´ee de fa¸con exponentielle. Abstract. We study algebraic properties of groups of PL or smooth homeomorphisms of unit cubes in any dimension, fixed pointwise on the boundary, and more generally PL or smooth groups acting on manifolds and fixing pointwise a submanifold of codimension 1 (resp. codimension 2), and show that such groups are locally indicable (resp. circularly orderable). We also give many examples of interesting groups that can act, and discuss some other algebraic constraints that such groups must satisfy, including the fact that a group of PL homeomorphisms of the n-cube (fixed pointwise on the boundary) contains no elements that are more than exponentially distorted. 1. Introduction We are concerned in this paper with algebraic properties of the group of PL homeomorphisms of a PL manifold, fixed on some PL submanifold (usually of codimension 1 or 2, for instance the boundary) and some of its subgroups (usually those preserving some structure). The most important case is the group of PL homeomorphisms of Infixed pointwise on ∂In; hence these are “groups of PL homeomorphisms of the (n-)cube”. The algebraic study of transformation groups (often in low dimension, or preserving some extra structure such as a symplectic or complex structure) has recently seen a lot of activity; however, much of this activity has been confined to the smooth category. It is striking that many of these results can be transplanted to the PL category. This interest is further strengthened by the possibility of working in the PL category over (real) algebraic rings or fields (this possibility has already been exploited in dimension 1, in the groups Fand Tof Richard Thompson). Many theorems we prove have analogs in the (C1) smooth category, and we usually give proofs of such theorems for comparison where they are not already available in the literature. Date: September 29, 2015. 1 2 DANNY CALEGARI AND DALE ROLFSEN 1.1. Statement of results. The main algebraic properties of our groups that we establish are left orderability (in fact, local indicability) and controlled distortion. These are algebraic properties which at the same time are readily compared with geometric or topological properties of a group action. For example, the property of left orderability for a countable group is equivalent to the existence of a faithful action on (I, ∂I) by homeomorphisms. As another example, control of (algebraic) distortion has recently been used by Hurtado [18] to prove strong rigidity results for homomorphisms between various (smooth) transformation groups. In §2 we state for the convenience of the reader standard definitions and results from the theory of left-orderable groups. In §3 we begin our analysis of PL groups of homeomorphisms of manifolds, fixed pointwise on a codimension 1 submanifold. If Mis a PL manifold and K a submanifold, we denote by PL+(M, K) the group of orientation-preserving PL homeomorphisms of Mfixed pointwise on K. We similarly denote by Diff+(M, K) and Homeo+(M, K) the groups of diffeomorphisms (resp. homeomorphisms) of a smooth (resp. topological) manifold M, fixed pointwise on a smooth (resp. topological) submanifold Kof codimension 1. The first main theorem of this section is: PL locally indicable Theorem 3.3.1. Let Mbe an ndimensional connected PL manifold, and let Kbe a nonempty closed PL submanifold of codimension at most 1. Then the group PL+(M, K)is left orderable; in fact, it is locally indicable. We give some basic constructions of interesting groups of PL homeomorphisms of cubes (of dimension at least 2) fixed pointwise on the boundary, including examples of free subgroups, groups with infinite dimensional spaces of (dynamically defined) quasimorphisms, and right-angled Artin groups; in fact, the method of Funar, together with a recent result of Kim and Koberda shows that every RAAG embeds in PL(I2, ∂I2). We conclude this section by studying distortion in PL(Im, ∂Im), and prove PL distortion Corollary 3.7.6. Every element of PL(Im, ∂Im)−Id is at most exponentially distorted. This lets us easily construct explicit examples of locally indicable groups which are not isomorphic to subgroups of PL(Im, ∂Im) for any m(for example, iterated HNN extensions). In §4 we prove Theorem 4.2.1, the analog of Theorem 3.3.1 for groups of C1 smooth diffeomorphisms. The main purpose of this section is to compare and contrast the methods of proof in the PL and smooth categories. In §5 we sharpen our focus to the question of bi-orderability (i.e. orders which are invariant under both left and right multiplication), for transformation groups acting on cubes in various dimensions and with differing degrees of analytic control. In dimensions bigger than 1, none of the groups we study are bi-orderable. In dimension 1, the groups PL(I, ∂I) and Diffω(I, ∂I) are bi-orderable. In §6 we consider groups acting on manifolds and fixing pointwise submanifolds (informally, “knots”) of codimension two. Here we are able to bootstrap our results from previous sections to show the following: PL circularly orderable Theorem 6.2.3. Let Mbe an ndimensional connected PL orientable manifold, and let Kbe a nonempty n−2dimensional closed submanifold. Then the group PL+(M, K)is circularly orderable. GROUPS OF PL HOMEOMORPHISMS OF CUBES 3 An analog for C1diffeomorphisms is also proved. An interesting application is to the case that Mis a 3-manifold, and Kis a hyperbolic knot (i.e. a knot with a hyperbolic complement). In this case, if G0(M, K) denotes the group of PL homeomorphisms (or C1diffeomorphisms) isotopic to the identity and taking K to itself (but not necessarily fixing it pointwise) then G0(M, K) is left-orderable. This depends on theorems of Hatcher and Ivanov on the topology of G0(M, K) as a topological group. Finally, in §7 we briefly discuss groups of homeomorphisms with no analytic restrictions. Our main point is how little is known in this generality in dimension >1; in particular, it is not even known if the group Homeo(I2, ∂I2) is left-orderable. We also make the observation that the groups Homeo(In, ∂In) are torsion-free for all n; this follows immediately from Smith theory, but the result does not seem to be well-known to people working on left-orderable groups, so we believe it is useful to include an argument here. 1.2. Acknowledgement. We would like to thank Andr´es Navas and Amie Wilkinson for some helpful discussions about this material. We are also very grateful to the anonymous referee who pointed us to several useful references. Danny Calegari was supported by NSF grant DMS 1005246. Dale Rolfsen was supported by a grant from the Canadian Natural Sciences and Engineering Research Council. 2. Left orderable groups This section contains a very brief summary of some standard results in the theory of left orderable groups. These results are collected here for the convenience of the reader. For proofs, see e.g. [7], Chapter 2. Definition 2.0.1 (Left orderable).A group Gis left orderable (usually abbreviated to LO) if there is a total order ≺on Gso that for all f, g, h ∈Gthe relation g≺h holds if and only if fg ≺fh holds. Lemma 2.0.2. If there is a short exact sequence 0→K→G→H→0and both Kand Hare left orderable, then Gis left orderable. Lemma 2.0.3. A group if left orderable if and only if every finitely generated subgroup is left orderable. Lemma 2.0.4. A countable group Gis left orderable if and only if it isomorphic to a subgroup of Homeo(I, ∂I). Definition 2.0.5 (Locally indicable).A group Gis locally indicable if for every finitely generated nontrivial subgroup Hof Gthere is a surjective homomorphism from Hto Z. Theorem 2.0.6 (Burns-Hale, [6]).Every locally indicable group is left orderable. 3. PL group actions on cubes 3.1. Definitions. We assume the reader is familiar with the concept of a PL homeomorphism between compact polyhedra in Rn, namely one for which the domain can be subdivided into finitely many linear simplices so that the restriction of the homeomorphism to each simplex is an affine linear homeomorphism to its image. We say that the simplices in the domain are linear for f. A PL manifold is one 4 DANNY CALEGARI AND DALE ROLFSEN with charts modeled on Rnand transition functions which are the restrictions of PL homeomorphisms. We define the dimension of a (linear) polyhedron to be the maximum of the dimensions of the simplices making it up, in any decomposition into simplices. Notation 3.1.1. Let Mbe a PL manifold (possibly with boundary) and let Kbe a closed PL submanifold of M. We denote by PL(M, K) the group of PL self-homeomorphisms of Mwhich are fixed pointwise on K. If Mis orientable, we denote by PL+(M, K) the subgroup of orientation-preserving PL selfhomeomorphisms. The main example of interest is the following: Notation 3.1.2. We denote by Inthe cube [−1,1]nin Euclidean space with its standard PL structure. Note with this convention that 0 is a point in the interior of In. We denote the boundary of Inby ∂In. So we denote by PL(In, ∂In) the group of PL self-homeomorphisms of the unit cube Inin Rnwhich are fixed pointwise on the boundary. If ωdenotes the standard (Lesbesgue) volume form on In, we denote by PLω(In, ∂In) the subgroup of PL(In, ∂In) preserving ω. 3.2. Transformation groups as discrete groups. Let Gbe a transformation group — i.e. a group of homeomorphisms of some topological space X. It is often useful to endow Gwith a topology compatible with the action; for instance, the compact-open topology. If we denote Gδas the same group but with the discrete topology, the identity homomorphism Gδ→Gis a continuous map of topological groups, and induces maps on cohomology H∗(BG;R)→H∗(BGδ;R) = H∗(K(G, 1); R) for any coefficient module R. Thus one interesting source of algebraic invariants of the discrete group Garise by thinking of its homotopy type as a topological group. However, the groups of most interest to us in this paper are not very interesting as (homotopy types of) topological spaces: Proposition 3.2.1 (Alexander trick).The groups Homeo(In, ∂In),PL(In, ∂In), PLω(In, ∂In)are all contractible in the compact-open topology. Proof. Given f:In→Infixed pointwise on ∂In, define ft:In→Inby ft(x) := (tf(x/t) if 0 ≤ kxk ≤ t xif t≤ kxk ≤ 1 where k · k denotes the L∞norm on In. Then f0=fand f1= Id, and the assignment f→ftdefines a deformation retraction of any one of the groups in question to the identity.  3.3. Left orderability. In this section we prove the left orderability of certain groups of PL homeomorphisms. The purpose of this section is to develop the tools to prove the following theorem: Theorem 3.3.1 (Locally indicable).Let Mbe an ndimensional connected PL manifold, and let Kbe a nonempty closed PL submanifold of codimension at most 1. Then the group PL+(M, K)is left orderable; in fact, it is locally indicable. GROUPS OF PL HOMEOMORPHISMS OF CUBES 5 An important special case is M=Inand K=∂In. First we introduce some notation and structure. Recall that if Xis a closed subset of Y, the frontier of Xin Yis the intersection of Xwith closure of Y−X. Notation 3.3.2. Let f∈PL(M, K). The fixed point set of fis denoted fix(f) and the frontier of fix(f) is denoted fro(f). Similarly, if Gis a subgroup of PL(M, K), the fixed point set of Gis denoted fix(G). The frontier of fix(G) is denoted fro(G). Lemma 3.3.3. Let Mbe an ndimensional connected PL manifold, and let Kbe a nonempty n−1dimensional closed PL submanifold. Let Gbe a nontrivial finitely generated subgroup of PL(M, K). (1) fix(G)is a closed polyhedron of dimension at least n−1. (2) fro(G)is a polyhedron of dimension n−1. (3) If g1,· · · , gmis a finite generating set for G, then fro(G)⊂ ∪ifro(gi). Proof. Let g1,· · · , gmbe a finite generating set for G. (1) We have fix(G) = ∩ifix(gi). Since each fix(gi) is a closed polyhedron containing K, so is fix(G). (2) If fix(G) has no interior, then its complement is open and dense, so fro(G) = fix(G). Otherwise, fix(G) has some interior, and fro(G) separates some point in the interior from some point in M−fix(G). Since Mis connected, fro(G) has dimension at least n−1. (3) Every point pin fro(G) is in fix(gi) for all i, and for some ithere are points arbitrarily near pmoved nontrivially by gi. Thus p∈fro(gi) for this i. Definition 3.3.4. Let Gbe a subgroup of PL(M, K). The action of Gis semilinear at a point pif there is some codimension 1 plane πthrough pand a convex open neighborhood Uof pso that the restriction of Gis linear on both components of U−(π∩U). We call πthe dividing plane. Note with this definition that a linear action at a point is semilinear. Note also that if Gis semilinear at a point pbut not linear there, the dividing plane is unique. Lemma 3.3.5. Let Mbe an ndimensional connected PL manifold, and let Kbe a nonempty n−1dimensional closed PL submanifold. Let Gbe a finitely generated subgroup of PL(M, K). Then for every n−1dimensional linear simplex σin fro(G), there is an open and dense subset of σwhere Gis semilinear, and the dividing plane is tangent to σ. Proof. Fix a finite generating set g1,· · · , gmfor G, and recall from Lemma 3.3.3 bullet (3) that fro(G)⊂ ∪ifro(gi). Let σbe an n−1 dimensional simplex in fro(G), and suppose σ∩fro(gi) has full dimension. Let σi=σ∩fro(gi). Associated to gi there is a decomposition of Minto linear simplices; away from the n−2 skeleton of this decomposition, giacts semilinearly. So giacts semilinearly on the complement of an n−2 dimensional polyhedral subset biof σi. For each point p∈σi−bithe action of gion the tangent plane to σiis the identity (since σiis fixed by gi) and therefore we can take the dividing plane to be equal to this tangent plane. For, either the action at pis semilinear but not linear (in which case any codimension 1 plane on which the action is linear is the unique dividing plane), or the action at pis linear, in which case any codimension 1 plane can be taking to be the dividing plane. Thus the dividing plane of giis tangent to σialong σi−bi. Furthermore, since 6 DANNY CALEGARI AND DALE ROLFSEN σ−σiis in fix(gi) but not fro(gi), the element giacts trivially in a neighborhood of each point of σ−σi. It follows that each giacts semilinearly at each point of σ− ∪ibiwith dividing plane tangent to σ, and therefore all of Gacts semilinearly at each point of σ−∪ibi with dividing plane tangent to σ. Lemma 3.3.6. The (pointwise) stabilizer in GL(n, R)of an mdimensional subspace πof Rnis isomorphic to Rm(n−m)oGL(m, R)and the subgroup preserving orientation is isomorphic to Rm(n−m)oGL+(m, R). Proof. The subgroup of GL(n, R) fixing πcan be conjugated into the set of matrices of the form Id V 0Awhere Id is the identity in GL(n−m, R), where Vis an arbitrary (n−m)×mmatrix, and where A∈GL(m, R). The proof follows.  Example 3.3.7 (Codimension 1 stabilizer).In the special case m=n−1 the stabilizer is isomorphic to Rn−1o R∗, and the orientation preserving subgroup is isomorphic to Rn−1o R+which is an extension of the locally indicable group R+ by the locally indicable group Rn−1, and is therefore locally indicable. We are now ready to give the proof of Theorem 3.3.1: Proof. By Theorem 2.0.6 it suffices to show that every nontrivial finitely generated subgroup of PL+(M, K) surjects onto Z. Let Gbe such a subgroup, and let pbe a point in a codimension 1 simplex in fro(G) where Gacts semilinearly. Since pis in fro(G), some giacts nontrivially on one side of fro(G), so there is a nontrivial homomorphism from Gto Rn−1o R+which is locally indicable, and therefore G surjects onto Z. Since Gwas arbitrary, the theorem is proved.  3.4. Free subgroups of PL(In, ∂In).In this section we discuss algebraic properties of the groups PL(In, ∂In) and their subgroups. The case n= 1 is well-studied, and one knows the following striking theorem: Theorem 3.4.1 (Brin-Squier, [5]).The group PL(I, ∂I)obeys no law, but does not contain a nonabelian free subgroup. On the other hand, we will shortly see that the group PL(In, ∂In) contains nonabelian free subgroups for all n > 1. Lemma 3.4.2. The group PL(I2, ∂I2)contains nonabelian free subgroups. Proof. For any M∈GL(2,R) there is some gMin PL(I2, ∂I2) which fixes 0, which is linear near 0, and which satisfies dgM(0) = M. Let Gbe the group generated by the gM. Then Gfixes 0 and is linear there, so there is a surjective homomorphism G→GL(2,R). If Fis a nonabelian free subgroup of GL(2,R), there is a section from Fto Gwhose image is a free subgroup of G. In fact we will shortly see (Theorem 3.6.3) that every right-angled Artin group embeds in PL(I2, ∂I2). Lemma 3.4.3. For all nthere is an injective homomorphism S: PL(In, ∂In)→PL(In+1, ∂In+1) called the suspension homomorphism. GROUPS OF PL HOMEOMORPHISMS OF CUBES 7 Proof. We include Ininto In+1 as the subset with last coordinate equal to 0. The bipyramid Pn+1 is the convex hull of Inand the two points (0,0,· · · ,0,±1). For every simplex σin Inthere are two simplices S±σof one dimension higher, obtained by coning σto (0,0,· · · ,0,±1). If fis a PL homeomorphism of Infixed on the boundary, and σis a simplex in Inlinear for f, then there is a unique PL homeomorphism Sf of Pn+1 which takes each S±σlinearly to S±f(σ). The map f→Sf defines an injective homomorphism PL(In, ∂In)→PL(Pn+1, ∂Pn+1). Then any element of PL(Pn+1, ∂P n+1) can be extended by the identity on In+1 − Pn+1 to an element of PL(In+1, ∂In+1).  Corollary 3.4.4. The group PL(In, ∂In)contains a nonabelian free subgroup for all n > 1. Proof. By Lemma 3.4.2, Lemma 3.4.3 and induction.  Actually, we will see many more constructions of free subgroups of PL(In, ∂In) in the sequel. 3.5. Area preserving subgroups. We are interested in the following subgroup of PL(I2, ∂I2). Definition 3.5.1. Let ωdenote the (standard) area form on I2. Let PLω(I2, ∂I2) denote the subgroup of PL(I2, ∂I2) consisting of transformations which preserve ω. The group PLω(I2, ∂I2) contains many interesting subgroups, and has a rich algebraic structure. We give some indications of this. Example 3.5.2 (Dehn twist).Figure 1 depicts an area-preserving PL homeomorphism of a square, which preserves the foliation by concentric squares, and (for suitable choices of edge lengths) whose 12th power is a Dehn twist (in fact, this transformation is contained in a 1-parameter subgroup of PLω(I2, ∂I2) consisting of powers of a Dehn twist at discrete times). −−−−−→ Figure 1. A 12th root of a PL Dehn twist The following theorem is due to Gratza [15]; in fact he proved the analogous statement for piecewise linear symplectic automorphisms of I2n. Theorem 3.5.3 (Gratza).The group PLω(I2, ∂I2)is dense in Diffω(I2, ∂I2), in the C0topology. 8 DANNY CALEGARI AND DALE ROLFSEN Gratza’s theorem has the following application: it can be used to certify that PLω(I2, ∂I2) admits an infinite dimensional space of nontrivial quasimorphisms (i.e. quasimorphisms which are not homomorphisms). By contrast, no subgroup of PL(I, ∂I) admits a nontrivial quasimorphism, by [8]. For an introduction to the theory of quasimorphisms and its relation to stable commutator length, see [9]. Definition 3.5.4. Let Gbe a group. A homogeneous quasimorphism on Gis a function φ:G→Rsatisfying the following properties: (1) (homogeneity) for any g∈Gand any n∈Zwe have φ(gn) = nφ(g); and (2) (quasimorphism) there is a least non-negative number D(φ) (called the defect) so that for all g, h ∈Gthere is an inequality |φ(gh)−φ(g)−φ(h)| ≤ D(φ) The (real vector) space of homogeneous quasimorphisms on Gis denoted Q(G). A function satisfying the second condition but not the first is said to be (simply) aquasimorphism. If φ:G→Ris any quasimorphism, the function ¯ φ:G→R defined by ¯ φ(g) = limn→∞ φ(gn)/n is a homogeneous quasimorphism, and satisfies ¯ φ−φ≤D(φ). This operation is called homogenization of quasimorphisms. A homogeneous quasimorphism has defect 0 if and only if it is a homomorphism to G. Thus Ddescends to a norm on the quotient space Q(G)/H1(G;R). It is a fact that Q/H1with the defect norm is a Banach space. It is known that Q/H1vanishes on any subgroup Gof PL(I, ∂I). By contrast, we show that the subgroup PLω(I2, ∂I2) admits an infinite dimensional Q/H1. The proof is by explicit construction, and based on a general method due to Gambaudo-Ghys [14]. The construction is as follows. First, fix some nand let Bn denote the braid group on nstrands, and fix ndistinct points x0 1,· · · , x0 nin the interior of I2. Let µ:Bn→Rbe any function. Now, since Homeo(I2, ∂I2) is path-connected and simply-connected, for any g∈ PLω(I2, ∂I2) there is a unique homotopy class of isotopy gtfrom gto Id. For any (generic) n-tuple of distinct points x1,· · · , xnin the interior of I2, and any g∈PLω(I2, ∂I2), let γ(g;x1,· · · , xn)∈Bnbe the braid obtained by first moving the x0 iin a straight line to the xi, then composing with the isotopy gtfrom xito g(xi), then finally moving the g(xi) in a straight line back to the x0 i(note that γdoes not depend on the choice of gt, since Homeo(I2, ∂I2) is simply-connected). Now we define Φµ(g) = ZI2×···×I2 µ(γ(g;x1,· · · , xn))dω(x1)dω(x2)· · · dω(xn) where the integral is taken over the subset of the product of ncopies of (the interior of) I2consisting of distinct n-tuples of points where γis well-defined. Lemma 3.5.5. Suppose µis a quasimorphism on Bn. Then Φµis a quasimorphism on PLω(I2, ∂I2) Proof. Changing the vector x0to a new vector y0changes γby multiplication by one of finitely many elements of Bn; since µis a quasimorphism, this changes Φµ by a bounded amount. For any two g, h ∈PLω(I2, ∂I2) and generic x1,· · · , xnwe have γ(gh;x1,· · · , xn) = γ(h;x1,· · · , xn)γ(g;h(x1),· · · , h(xn)) GROUPS OF PL HOMEOMORPHISMS OF CUBES 9 Now integrate over I2× · · · × I2and use the fact that µis a quasimorphism to see that Φµis a quasimorphism.  The homogenization Φµis a homogeneous quasimorphism, and one may check that it is nontrivial whenever µis nontrivial on Bn. In fact, it is easiest to check this for the special case of homogeneous quasimorphisms on Bnthat vanish on reducible elements. Such quasimorphisms are plentiful; for example, any “counting quasimorphism” arising from a weakly properly discontinuous action of Bnon a hyperbolic simplicial complex. See [9], §3.6 for more details. Lemma 3.5.6. Let µbe nontrivial on Bnand vanish on all reducible elements. Then Φµis nontrivial on PLω(I2, ∂I2). Proof. For 1 ≤i≤nlet Ribe the rectangle with lower left corner (i−1 n+, ) and upper right corner ( i n−, 1−). For any braid b∈Bnwe can build an areapreserving homeomorphism φbwhich permutes the Ri, taking each Rito its image by a translation, and which performs the conjugacy class of the braid bon each n tuple of points of the form (x, x + 1,· · · , x +n−1) for x∈R1. If we are systematic about the way we extend φbto I2−∪iRi(i.e. by decomposing φbinto a product of standard “elementary” moves, corresponding to the factorization of binto elementary braids) we can ensure that φbsatisfies an estimate of the form |µ(γ(φb;x1,· · · , xn))| ≤ C· |b| · D(µ) where |b|denotes the word length of b, where D(µ) denotes the defect of µ, and where Cis some constant depending only on n, but not on . It is straightforward to build such an area-preserving homeomorphism; to see that it can be approximated by a PL area-preserving homeomorphism with similar properties, we appeal to Gratza’s Theorem 3.5.3. For any n-tuple x:= (x1,· · · , xn) where the xiare all contained in distinct Rj, the powers of φbon xare conjugates of the powers of b. For n-tuples where two xi are in the same Rj, the powers of φbon xare reducible. Thus we can estimate Φµ(φb) = (n!/nn)µ(b) + O() Taking bto be a braid on which µis nonzero, and taking →0 we obtain the desired result.  In particular, Q(PLω(I2, ∂I2)) is infinite dimensional. This should be contrasted with the 1-dimensional case, where it is shown in [8] that for any subgroup Gof PL(I, ∂I) the natural map H1(G)→Q(G) is surjective. 3.6. Right-angled Artin groups. Recall that a right-angled Artin group (hereafter a RAAG) associated to a (finite simplicial) graph Γ is the group with one generator for each vertex of Γ, and with the relation that two generators commute if the corresponding vertices are joined by an edge, and with no other relations. We denote this group by A(Γ) Funar [13] discovered a powerful method to embed certain RAAGs in transformation groups. To describe Funar’s method, it is convenient to work with the complement graph Γc, which has the same vertex set as Γ, and which has an edge between two distinct vertices if and only if Γ does not have an edge between these vertices. Thus: two vertices of Γcare joined by an edge if and only if the corresponding generators of the RAAG do not commute. Let Sbe a surface, and for each vertex iof Γc, let γibe an embedded circle in S, chosen with the following properties: 16 DANNY CALEGARI AND DALE ROLFSEN Proof. Let pi→pand suppose that Ghas fixed rank ≥mon all pi. If m=−1 there is nothing to prove. Otherwise, the piare all fixed by every g∈Gand there is a subspace πi⊂Tpiof dimension mfixed by every dg|pi. By compactness of the Grassmannian of m-dimensional subspaces of Rn, after passing to a subsequence we can assume that πiconverges to some m-dimensional subspace π⊂Tp. Observe that for every gwe must have that dg|pfixes π, or else some dg|piwould fail to fix πi, since the action is C1. Thus the fixed rank of Gat pis ≥m. We can now give the proof of Theorem 4.2.1: Proof. Let Gbe a nontrivial finitely generated subgroup of Diff1 +(M, K), and let X be the subset of Gwhere the fixed rank is ≥n−1. Since Gis arbitrary, it suffices to show that Gadmits a surjection to Z. By Lemma 4.2.3 the set Xis closed. Furthermore, it is nonempty, since it includes K. Since Gis nontrivial, some point of Mis moved by some element of G, and therefore the fixed rank is −1 at that point; since Mis connected, it follows that the frontier fro(X) is nonempty. Let p∈fro(X) be arbitrary, and let π⊂TpMbe an n−1 dimensional plane fixed by dg|pfor all g∈G. Let ρ:G→GL(n, R) send g→dg|p. By Lemma 3.3.6 and Example 3.3.7 the image of ρis locally indicable. If the image is nontrivial we are done. So suppose the image is trivial. Since pis in the frontier of Xit must be in the frontier of fix(G) (or else it would be in the interior of the set where the fixed rank is n) so the image of Gin the group of germs of diffeomorphisms fixing pis nontrivial. So the Thurston stability theorem (i.e. Theorem 4.1.2) implies that Gsurjects to Z. This completes the proof.  5. Bi-orderability If a left-order ≺of a group Gis also invariant under right-multiplication, we’ll call it a bi-order and say that Gis bi-orderable. It is easy to see that a left-order ≺ is a bi-order if and only if its positive cone P:= {g∈Gsuch that 1 ≺g} is invariant under conjugation; i.e. if and only if g−1P g ⊂Pfor all g∈G. In this section we discuss bi-orderability for various subgroups of Homeo(In, ∂In) in specific dimensions, and with various kinds of regularity. 5.1. Bi-orderability in general. First of all, it should be noted that the distinction between left orderability and bi-orderability is not vacuous: Example 5.1.1.If Gis a group in which there is a nontrivial element gso that some product of conjugates of gis trivial, then Gis not bi-orderable. For, in any left ordering, we may assume that g∈P(up to reversing the order). So, for example, the Klein bottle group ha, b |aba−1=b−1iis not bi-orderable, though it is locally indicable. One may summarize the existence of LO groups which are not bi-orderable in the following proposition: Proposition 5.1.2. Homeo(I, ∂I)is not bi-orderable. This is because Homeo(I, ∂I) contains isomorphic copies of all countable LO groups, many of which are not bi-orderable. Suspending to higher dimensions, we conclude: GROUPS OF PL HOMEOMORPHISMS OF CUBES 17 Corollary 5.1.3. For all n≥1,Homeo(In, ∂In)is not bi-orderable. 5.2. Bi-orderability in PL. The following was observed in a slightly different context by Chehata [11]. Proposition 5.2.1. PL(I, ∂I)is bi-orderable. One can take as positive cone Pfor a bi-ordering all PL functions f:I→I whose graph departs from the diagonal for the first time with slope greater than 1. In other words, if x0is the maximal element of I= [−1,1] such that f(x) = xfor all x∈[−1, x0], then for all sufficiently small  > 0 we have f(x0+)> x0+. Then Pis closed under composition of functions, conjugation, and PL(I, ∂I) = {1}tPtP−1, so it defines a bi-order by the recipe g≺h⇐⇒ g−1h∈P. Proposition 5.2.2. PLω(I2, ∂I2)is not bi-orderable. To see this, we consider two functions f, g ∈PLω(I2, ∂I2) defined as follows. Let hdenote the function illustrated in Figure 1. Recall that h12 is a Dehn twist, which is the identity on the inner square, as well as on ∂I2. Let f:= h6, so that frotates the inner square by 180 degrees. Referring to Figure 3, define gto be the identity outside the little squares, which are strictly inside the inner square rotated by f. On the little square on the left, let gact as h, suitably scaled, and on the little square to the right let gact as h−1. Noting that finterchanges the little squares, and that hcommutes with 180 degree rotation, one checks that f−1gf =g−1. Such an equation cannot hold (for gnot the identity) in a bi-ordered group, as it would imply the contradiction that gis greater than the identity if and only if g−1 is greater than the identity (this is just the Klein bottle group from Example 5.1.1). hh−1 Id Figure 3. Building the function g∈PLω(I2, ∂I2) Corollary 5.2.3. For n≥2none of the groups PLω(In, ∂In),PL(In, ∂In)is biorderable. This follows from Lemma 3.4.3, noting that suspension also takes PLω(In, ∂In) isomorphically into PLω(In+1, ∂In+1). Recall Theorem 3.3.1: if Mis a connected PL n-manifold and Kis a nonempty closed PL n−1 dimensional submanifold, then PL(M, K) is LO. Noting that one may include PL(In, ∂In) in PL(M, K), by acting on a small n-ball in M, we see also 18 DANNY CALEGARI AND DALE ROLFSEN Corollary 5.2.4. For n≥2,PL(M, K)is not bi-orderable. 5.3. Bi-orderability in Diff. By an argument similar to Proposition 5.2.2 one can show Proposition 5.3.1. For n≥2and for any 0≤p≤ ∞, the group Diffp(In, ∂In) is not bi-orderable. Note that in every case bi-orderability is ruled out by the existence of a nontrivial element which is conjugate to its inverse. For n= 1 and p= 0,1 bi-orderability in Diffp(I, ∂I) is ruled out for the same algebraic reason: Example 5.3.2 (C1example).For i∈Zlet xibe a discrete, ordered subset of the interior of Iaccumulating at the endpoints like the harmonic series. Let fbe a C1 diffeomorphism whose fixed point set in the interior of Iis exactly the union of the xi, and which alternates between translating in the positive and negative direction on intervals (xi, xi+1) for irespectively even and odd. Let gtake xito xi+1 and conjugate the action of fto f−1. This construction can evidently be made C1in the interior. Moreover, if we arrange for f0to converge uniformly to 0 towards the endpoints, then the same is true of g, since gacts there almost like a linear map [1 −1/i, 1] →[1 −1/(i+ 1),1], whose derivatives converge uniformly to 0. On the other hand, this construction cannot be made C2, by Kopell’s Lemma, which says the following: Theorem 5.3.3 (Kopell’s Lemma [22], Lemma 1).Let fand gbe commuting elements of Diff2(I, ∂I). If ghas no fixed point in the interior of Ibut fhas a fixed point in the interior of Ithen f= Id. Corollary 5.3.4. In Diff2(I, ∂I)no nontrivial element is conjugate to its inverse. Proof. If fis nontrivial and conjugate to its inverse, there are some subintervals where it acts as a positive translation, and some where it acts as a negative translation, and some point between the two is fixed. Let gconjugate fto f−1. Applying Kopell’s lemma to gand f2we see that f2= Id so f= Id.  In particular, the Klein bottle group does not embed in Diff2(I, ∂I). Remark 5.3.5.We suspect that the group Diffp(I, ∂I) is not bi-orderable for any finite p, but have not been able to show this. In this remark we give a detailed construction of a Cpgroup action (for any fixed p) containing two specific elements which are C1conjugate; if the construction could be done in such a way that the elements are Cpconjugate, it would prove that Diffp(I, ∂I) is not bi-orderable, but we have not been able to show this. We build a Cpdiffeomorphism fof Ias follows (for simplicity of formulae, we take here Ito be the interval [0,1] instead of [−1,1] as throughout the rest of the paper). For i < 0 let pi= 2iand for i≥0 let pi= 1−2−i−2. We build ffixing each pi, and acting alternately as a positive and negative translation on complementary regions. We will insist that fis infinitely tangent to the identity at each pi. For concreteness, on each of the intervals [7/8,15/16] and [1/16,1/8] we let fbe the time 1 flow of a C∞vector field with no zeros in the interior of the intervals, and infinitely tangent to zero at the endpoints. The definition of fon the rest of Iwill be defined implicitly in what follows. GROUPS OF PL HOMEOMORPHISMS OF CUBES 19 Now, we let gbe a diffeomorphism which takes pito pi+1 for i≥0 and takes pito pi−1for i < 0, and let gact as a dilation (i.e. with locally constant nonzero derivative) on open neighborhoods of 0 and 1. Evidently we can choose such a gto be C∞. Finally, we arrange the dynamics of fon the complementary intervals so that gconjugates fto f−koutside the interval [1/4,7/8] for some big k, depending on p(which we will specify shortly). Since f(on the intervals [7/8,15/16] and [1/16,1/8]) is the time 1 flow of a vector field, taking an nth root multiplies each Cp norm by O(1/n) when nis big. On the other hand, conjugating a diffeomorphism by dilation by 1/λ blows up the Cpnorm by a factor of λp−1, so providing k > 2p−1 the diffeomorphism fwe build with this property will be C∞on (0,1), and Cp tangent to the identity at 0 and 1, and therefore is an element of Diffp(I, ∂I). By construction, the product h:= fkgfg−1is supported on the interval [1/4,7/8] which is subdivided into 3 intervals J0, J1, J2where hacts alternately as positive, negative, positive translation. If we are careful in our choice of g, we may ensure that his infinitely tangent to the identity at the endpoints of the Ji. By a similar construction, we can choose a slightly different element j, whose germs at 0 and 1 agree with g, so that jconjugates fto f−koutside the interval [1/8,3/4], and then we can arrange that h0:= fkjfj−1is supported on the interval [1/8,3/4] and acts alternately on a subdivision J0 0, J0 1, J0 2as negative, positive, negative translation, and infinitely tangent to the identity at the endpoints. Now, suppose for a suitable choice of gand jas above, we could find some e which conjugates h0to h−1, so that we get a relation of the form e(fkjfj−1)e−1fkgfg−1= Id But this word is a product of conjugates of f, and therefore a relation of this kind would certify that Diffp(I, ∂I) is not bi-orderable. We suspect that such an ecan be found (for some gand j), but leave this as an open problem. In view of the construction outlined in Remark 5.3.5 it is a bit surprising that the group Diffω(I, ∂I)is bi-orderable: Proposition 5.3.6. The group Diffω(I, ∂I)is bi-orderable. Proof. Define a bi-ordering on Diffω(I, ∂I) as follows. If the derivative f0(0) is >1, put the element fin the positive cone. If f0(0) = 1, look at the Taylor expansion at 0. This is of the form f(t) = t+a2t2+a3t3+· · · Since fis real-analytic, either f= Id, or else there is some first index isuch that ai6= 0. Then put fin the positive cone if ai>0. One checks that this really does define a cone, and that for every nontrivial element f, either fis in the cone, or f−1is in the cone. Finally, the cone is conjugation invariant. More geometrically, fis in the positive cone if f(t)> t for all sufficiently small t > 0. That this is conjugation invariant and a cone is obvious. That it is welldefined follows from the fact that a nontrivial real analytic diffeomorphism of I can’t have infinitely many fixed points.  We remark Proposition 5.3.6 has already appeared in the literature; see [24], Example 3.5. 20 DANNY CALEGARI AND DALE ROLFSEN 6. Groups fixing codimension 2 submanifolds In this section we establish analogs of Theorem 3.3.1 and Theorem 4.2.1 for groups acting in a PL or smooth manner on manifolds, fixing a submanifold of codimension 2 pointwise. The conclusion is not that such groups are left orderable (they are not in general) but that they are circularly orderable. 6.1. Circularly ordered groups. We recall standard definitions and facts about circular orders; [7], Chapter 2 is a reference. Definition 6.1.1. Let Σ be a set and let Σ(3) denote the space of distinct. ordered triples in Σ. A circular order on Σ is a map e: Σ(3) → ±1 satisfying the cocycle condition e(s1, s2, s3)−e(s0, s2, s3) + e(s0, s1, s3)−e(s0, s1, s2) = 0 for all distinct quadruples s0, s1, s2, s3in Σ. A (left) circular order on a group Gis a circular order on Gwhich is invariant under left-multiplication; i.e. it satisfies e(gg1, gg2, gg3) = e(g1, g2, g3) for all gin Gand all distinct triples (g1, g2, g3)∈G(3). Remark 6.1.2.Note that emay be extended to all triples Σ3by defining eto be 0 on Σ3−Σ(3); this extension also satisfies the cocycle condition. Notation 6.1.3. A group which admits an invariant circular order is said to be circularly orderable, which we usually abbreviate by CO. Lemma 6.1.4. Let Gbe a group. Suppose every finitely generated subgroup of G is CO. Then Gis CO. Lemma 6.1.5. If there is a short exact sequence 0→K→G→H→0where H is circularly orderable and Kis left orderable, then Gis circularly orderable. In particular, left orderable groups are circularly orderable. Lemma 6.1.6. A countable group is circularly orderable if and only if it is isomorphic to a subgroup of Homeo+(S1). Conversely, every subgroup of Homeo+(S1)is circularly orderable. Lemma 6.1.7. Suppose Gis circularly orderable. The (extended) function eis a Z-valued 2-cocycle on Gand thereby determines a class [e]∈H2(G;Z). If the class [e] = 0 then Gis left orderable. Example 6.1.8.The group GL+(2,R) is an extension 0 →R+→GL+(2,R)→ SL(2,R)→0. The group SL(2,R) acts faithfully and in an orientation-preserving way on the circle of linear rays through the origin, and is therefore circularly orderable, by Lemma 6.1.6. By Lemma 6.1.5 the group GL+(2,R) is CO. 6.2. Knots. We would like to extend the results from §3.3 and §4.2 to pairs (M, K) where the codimension of Kis 2. Motivated by the interesting example where M=S3and Kis a circle, we use the informal term “knot” for Kalthough we do not assume either that Kis connected, or that it is homeomorphic to a sphere. GROUPS OF PL HOMEOMORPHISMS OF CUBES 21 Theorem 6.2.1 (C1Knots CO).Let Mbe an ndimensional connected smooth orientable manifold, and let Kbe a nonempty n−2dimensional closed submanifold. Then the group Diff1 +(M, K)is circularly orderable. Proof. Fix some point k∈K. The linear representation ρ: Diff1 +(M, K)→ GL(n, R) defined by ρ(g) = dg|kfixes a subspace πof dimension n−2, and therefore by Lemma 3.3.6 has image in R2(n−2) oGL+(2,R). The image group is CO by Lemma 6.1.5, since GL+(2,R) is CO (by Example 6.1.8) and R2(n−2) is locally indicable and therefore LO. Let Kdenote the kernel, and let Gbe a finitely generated subgroup of the kernel. If the germ of Gat kis nontrivial, then Gsurjects onto Z, by the Thurston stability theorem (i.e. Theorem 4.1.2). If the germ of G at kis trivial, then each of the finite generators gifixes an open neighborhood of k, and therefore fix(G) has nonempty interior. It follows that the subset Xwhere the fixed rank of Gis ≥n−1 is nonempty, and since Mis connected fro(X) is also nonempty. Therefore as in the proof of Theorem 4.2.1 we deduce that Gsurjects onto Z. It follows that Kis locally indicable and therefore LO, so Diff1 +(M, K) is CO by Lemma 6.1.5.  Before proving the corresponding theorem for PL actions, we must analyze the local structure of a group of PL homeomorphisms at an arbitrary point on an m-dimensional fixed submanifold. Lemma 6.2.2. Let Gbe a countable group of germs at 0of PL diffeomorphisms of Rnfixing Rm. There is a homomorphism from Gto the group of piecewise projective automorphisms of RPn−m−1, and for every finitely generated subgroup Hof the kernel, either Hsurjects to Zor fix(H)has nonempty interior arbitrarily close to 0. In particular, the kernel is locally indicable, and therefore LO. Proof. A linear automorphism of Rnfixing Rmacts linearly on the quotient Rn/Rm and therefore projectively on RPn−m−1. For each element g∈Gwe can pick a subdivision into linear simplices such that the restriction of gto each simplex is linear. The set of points on Rmin the complement of the m−1 skeleton of this subdivision is open and dense, and therefore since Gis countable, there is a point pon Rmnear 0 which is in the complement of the m−1 skeleton of the subdivision associated to every g∈G. For every g∈Gand every simplex σin the linear subdivision associated to g, there is a well-defined projective action of gon each simplex of the link of g. Putting these actions together for all g∈Ggives a piecewise projective action of Gon RPn−m−1at p. A finitely generated subgroup Hof the kernel locally preserves the foliation by planes parallel to Rm, and acts there by translation. If this action is trivial, fix(H) has interior near p. Theorem 6.2.3 (PL Knots CO).Let Mbe an ndimensional connected PL orientable manifold, and let Kbe a nonempty n−2dimensional closed submanifold. Then the group PL+(M, K)is circularly orderable. Proof. Let Gbe a countable subgroup of PL+(M, K) and consider the action near a point pin an n−2 dimensional simplex in K. By Lemma 6.2.2 there is a homomorphism from the germ of Gat pto the group of piecewise projective automorphisms of RP1with LO kernel. Thus the germ of Gat pis CO. Any finitely generated 22 DANNY CALEGARI AND DALE ROLFSEN subgroup Hof the kernel of the map from Gto the germ at phas the property that fix(H) has nonempty interior, and therefore His LO (in fact, locally indicable) as in the proof of Theorem 3.3.1. Thus every countable subgroup of PL+(M, K) is CO, and therefore PL+(M, K) is CO by Lemma 6.1.4.  6.3. Hyperbolic knots. An interesting application is to groups acting on a 3manifold stabilizing a knot. Explicitly, let’s specialize to the case that Mis an orientable 3-manifold, and Kis a knot with hyperbolic complement. Let G(M, K) denote the group of orientation preserving PL (resp. Diff1) homeomorphisms of M that take Kto itself by an orientation preserving homeomorphism, where we do not assume Kis fixed pointwise; and let G0(M, K) denote the subgroup of such transformations isotopic to the identity. Since M−Kis Haken, as a topological group, the homotopy type of G(M, K) is well-understood by the work of Hatcher and Ivanov. In fact, one has: Theorem 6.3.1 (Hatcher; Hatcher, Ivanov).As a topological group, G0(M, K)is contractible. The PL case is due independently to Hatcher and Ivanov; see e.g. [16] or [19]. As was well-known at the time, the smooth case follows from the PL case once one knows the Smale conjecture, proved by Hatcher [17]. Corollary 6.3.2. With notation as above, G0(M, K)is left orderable. Proof. Let’s abbreviate G0(M, K) by Gin what follows. The image of Gin Homeo+(K) is circularly orderable, and we have already shown that the kernel is circularly orderable. We would like to show that both the image and the kernel are actually LO. By Lemma 6.1.7 we need to show that the cohomology class [e] associated to either circular order is trivial. Notice by construction that [e] is an element of H2(BG;Z) where Gis thought of as a topological group, and BG denotes the classifying space for principle G-bundles. But since M−Kis hyperbolic, Gis contractible, and therefore H2(G;Z) = 0.  7. Groups of homeomorphisms It is natural to wonder whether the groups Homeo(M, K) are leftor circularlyordered when Kis of codimension 1 or 2, by analogy with the PL or smooth case. But here the situation is utterly mysterious, even in dimension 2. In fact, the following question still seems far beyond reach: Question 7.0.3. Is Homeo(I2, ∂I2)left-orderable? It is challenging to obtain any restrictions on the subgroups of Homeo(In, ∂In) at all, for n≥2. The purpose of this section is to point out that Smith theory shows that the groups Homeo(In, ∂In) are torsion free. This fact seems to be well known to the experts in Smith theory and its generalizations, but not to people working on left-orderable groups, and therefore it seems worthwhile to give a proof (or rather to point out how the result follows immediately from results which are well-documented in the literature). We use the following theorem, generalizing work of Smith, and due in this generality to Borel [4]: GROUPS OF PL HOMEOMORPHISMS OF CUBES 23 Theorem 7.0.4 (Smith, Borel).Let Xbe a finitistic space (e.g. a compact space) with the mod phomology of a point, and let Gbe a finite p-group acting on Xby homeomorphisms. Then XG(the fixed point set of G) has the mod phomology of a point. Here a finitistic space is one such that every covering has a finite dimensional refinement. Any compact space is finitistic, so Inis certainly finitistic (and the main application of Smith theory is to manifolds and manifold-like spaces). Corollary 7.0.5. The group Homeo(In, ∂In)is torsion-free for all n. 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