Proper 2–equivalences between infinite ended finitely presented groups

A T G Algebraic & Geometric Topology msp Volume 23 (2023) Proper 2–equivalences between infinite ended finitely presented groups MANUEL CÁRDENAS FRANCISCO FERNÁNDEZ LASHERAS ANTONIO QUINTERO RANJA ROY msp Algebraic & Geometric Topology 23:1 (2023) 1–11 DOI: 10.2140/agt.2023.23.1 Published: 27 March 2023 Proper 2–equivalences between infinite ended finitely presented groups MANUEL CÁRDENAS FRANCISCO FERNÁNDEZ LASHERAS ANTONIO QUINTERO RANJA ROY Recall that two finitely presented groups G and H are “proper 2 –equivalent” if they can be realized by finite 2 –dimensional CW–complexes whose universal covers are proper 2 –equivalent as (strongly) locally finite CW–complexes. This purely topological relation is coarser than the quasi-isometry relation, and those groups which are 1 –ended and semistable at infinity are classified, up to proper 2 –equivalence, by their fundamental pro-group. We show that if G and H are proper 2 –equivalent and semistable at each end, then any two finite graph of groups decompositions of G and H with finite edge groups and finitely presented vertex groups with at most one end must have the same set of proper 2 –equivalence classes of (infinite) nonsimply connected at infinity vertex groups (without multiplicities). Moreover, those simply connected at infinity vertex groups in such a decomposition (if any) are all proper 2 –equivalent to ZZZ . Thus, under the semistability hypothesis, this answers a question concerning the classification of infinite ended finitely presented groups up to proper 2 –equivalence, and shows again the behavior of proper 2 –equivalences versus quasi-isometries, in which the geometry of the group is taken into account. 57M07; 57M10 1 Introduction In [5], Gromov outlined a program to understand and try to classify all finitely generated groups geometrically via the notion of quasi-isometry, regarded as metric spaces. Since then, those properties of finitely generated groups which are invariant under quasiisometries have been of great interest and widely studied. On the other hand, the study of asymptotic invariants of a topological nature for finitely generated groups has also led to an interesting research area; see Geoghegan [4] for a good source on this subject. © 2023 MSP (Mathematical Sciences Publishers). Distributed under the Creative Commons Attribution License 4.0 (CC BY). Open Access made possible by subscribing institutions via Subscribe to Open. 2Manuel Cárdenas, Francisco Fernández Lasheras, Antonio Quintero and Ranja Roy In this realm, in Cárdenas, Lasheras, Quintero and Roy [2] a topological equivalence relation was introduced within the class of finitely presented groups attending to their asymptotic topology rather than their asymptotic geometry. More precisely, two finitely presented groups G and H are said to be proper 2 –equivalent if there exist (equivalently, for all) finite 2 –dimensional CW–complexes X and Y , with 1.X/ ŠG and 1.Y / ŠH , such that their universal covers z X and z Y are proper 2 –equivalent (as locally finite CW–complexes); in fact, the required proper 2 –equivalence can be replaced by a proper homotopy equivalence after wedging with 2 –spheres. It is worth pointing out that this equivalence relation is coarser than the quasi-isometry relation; quasi-isometric finitely presented groups are also related in this wider and “geometry forgetful” sense, by [4, Theorem 18.2.11]. It has been shown that two finite graph of groups decompositions with finite edge groups and finitely presented vertex groups with at most one end yield proper 2 –equivalent groups if they have the same set of proper 2 –equivalence classes of vertex groups; see [2, Theorem 3.9]. On the other hand, in contrast to the situation under the quasiisometry relation (see Papasoglu and Whyte [14, Theorem 0.4]), the converse does not hold in general. The proper 2 –equivalence class of a finitely presented group does not determine in general the set of proper 2 –equivalence classes of vertex groups in such a decomposition of the group; see [2]. We establish a partial converse to [2, Theorem 3.9] under the semistability hypothesis: Theorem 1.1 Let G and H be two proper 2 –equivalent finitely presented groups which are semistable at each end , and let .G; / and .H; 0/ be finite graph of groups decompositions of G and H with finite edge groups and finitely presented vertex groups with at most one end. Then ,.G; / and .H; 0/ have the same set of proper 2 – equivalence classes of ( infinite ) nonsimply connected at infinity vertex groups ( without multiplicities ) . Moreover , those simply connected at infinity vertex groups in .G; / and .H; 0/(if any)are all proper 2–equivalent to ZZZ. Observe that, by [14, Theorem 0.4], if G and H are in fact quasi-isometric then .G; / and .H; 0/ have the same set of proper 2 –equivalence classes of vertex groups (without multiplicities), regardless of the semistability hypothesis. Remark 1.2 The statement of Theorem 1.1 is the best possible, as the example in [2, Section 6]shows that GDZ2Z2Z2 and HD.ZZZ/.ZZZ/ are proper 2 –equivalent, but the vertex groups in the given graph of groups decompositions are all either finite in the case of Gor simply connected at infinity in the case of H. Algebraic & Geometric Topology, Volume 23 (2023) Proper 2–equivalences between infinite ended finitely presented groups 3 2 Some preliminaries For the most part we will be working within the category of locally finite CW–complexes and proper maps. We recall that a proper map is a map with the property that the inverse image of every compact subset is compact. Thus, two locally finite CW–complexes are said to be proper homotopy equivalent if they are homotopy equivalent and all homotopies involved are proper. On the other hand, a proper cellular map fWX!Y between finite-dimensional locally finite CW–complexes is a proper n –equivalence if there is another proper cellular map gWY!X such that the restrictions gıfjXn1 and fıgjYn1 are proper homotopic to the inclusion maps Xn1X and Yn1Y . Observe that if two finite-dimensional locally finite CW–complexes are proper homotopy equivalent then they are proper n –equivalent, for all n . Also, one can easily check that if two finite-dimensional locally finite CW–complexes are proper n –equivalent then so are their n –skeleta, by the proper cellular approximation theorem [4, Theorem 10.1.14]. Given a noncompact (strongly) locally finite CW–complex Y , a proper ray in Y is a proper map !WŒ0; 1/!Y . We say that two proper rays ! and !0 define the same end if their restrictions to the natural numbers !jN and !0jN are properly homotopic. This equivalence relation gives rise to the notion of the end determined by ! as the corresponding equivalence class, as well as the space of ends E.Y / of Y as a compact totally disconnected metrizable space; see [4, Section 13.4]. The CW–complex Y is semistable at the end determined by ! if any other proper ray defining the same end is in fact properly homotopic to ! . Equivalently, Y is semistable if the fundamental pro-group pro- 1.Y; !/ is pro-isomorphic to a tower of groups with surjective bonding homomorphisms; see [4, Proposition 16.1.2]. Recall that pro- 1.Y; !/ is represented by the inverse sequence (tower) of groups 1.Y; !.0// 1  1.Y C1; !.t1// 2  1.Y C2; !.t2//    ; where C1C2     Y with !.Œti;1// YCi is a filtration of Y by compact subspaces, and the bonding homomorphisms i are induced by the inclusions and basepoint-change isomorphisms (which are defined using subpaths of ! ). One can show the independence with respect to the filtration. Also, properly homotopic base rays yield pro-isomorphic fundamental pro-groups. We refer to [4;9] for more details and the basics of the pro-category of towers of groups. Given a CW–complex X with 1.X/ ŠG we will denote by z X the universal cover of X , constructed as prescribed in [4, Section 3.2], so that G acts freely on the CW– complex z X via a cell-permuting left action with Gnz XDX . The number of ends of an Algebraic & Geometric Topology, Volume 23 (2023) 4Manuel Cárdenas, Francisco Fernández Lasheras, Antonio Quintero and Ranja Roy (infinite) finitely generated group G represents the number of ends of the (strongly) locally finite CW–complex z X1 , for some (equivalently any) CW–complex X with 1.X/ ŠG and with finite 1 –skeleton, which is either 1 , 2 or 1 (finite groups have 0 ends [4;15] ). If G is finitely presented, then G is semistable at each end (or at infinity, if G is 1 –ended) if the (strongly) locally finite CW–complex z X2 is so, for some (equivalently any) CW–complex X with 1.X/ ŠG and with finite 2 –skeleton. In fact, we will refer to the fundamental pro-group of z X2 (at each end) as the fundamental pro-group of G (at each end), and will say that G is simply connected at each end (or at infinity, if G is 1 –ended) if it has pro-trivial fundamental pro-group at each end. Observe that any finite-dimensional locally finite CW–complex is strongly locally finite; see [4, Proposition 10.1.12]. 3 Proof of Theorem 1.1 We need the following result, which can be seen as a generalization of [12, Lemma 5] : Proposition 3.1 Let G be a finitely presented group which is semistable at each end , and assume G splits as an amalgamated product G0FG1( resp. an HNN-extension HF) over a finite group F . Then each factor G0 and G1( resp. the base group H) is either finite or else it is also semistable at each end. Moreover , if the fundamental pro-group of G at a certain end is nontrivial then it is pro-isomorphic to the fundamental pro-group of one of the factors (resp. the base group)at one of its ends. The proof of the first part of this proposition mimics that of [7, Lemma 3.2]. Nonetheless, we include a detailed proof for the sake of completeness. It is worth mentioning that independently and essentially simultaneously, Mihalik has recently shown a similar result; see [11, Theorem 3.3]. Proof Let G0 and G1 be finitely presented groups and F be a finite group with presentation he1; : : : ; enIr1; : : : ; rmi . Consider monomorphisms 'iWF!Gi for iD0; 1 , and denote by GDG0FG1D hG0; G1I'0.ei/D'1.ei/; 1 ini the corresponding amalgamated product. Let K0 and K1 be finite 2 –dimensional CW–complexes with 1.Ki/ŠGi , and let fiWWn iD1S1!Ki.i D0; 1/ be cellular maps such that Im fi1.Ki/ corresponds to the subgroup Im 'iGi . Let L0 be the standard 2 –complex associated to the given presentation of F , with 1 –cells e1; : : : ; en . Namely, L0 is a wedge of circles each of which is directed and labeled by one of the Algebraic & Geometric Topology, Volume 23 (2023) Proper 2–equivalences between infinite ended finitely presented groups 5 generators, and a 2 –cell attached to the 1 –skeleton according to each of the defining relations. Consider the adjunction spaces LDWn iD1eiI[.Wn iD1ei/f1 2gL0˚1 2 (homotopy equivalent to L0 ) and XDL[f0f0g[f1f1g.K0tK1/ . By van Kampen’s theorem, X is a finite 2 –dimensional CW–complex with 1.X/ ŠG0FG1 . Let z X be the universal cover of X with covering map pWz X!X . Observe that p1.Ki/ consists of a disjoint union of copies of the universal cover z Ki of Ki , since the inclusion Ki,!X induces a monomorphism Gi,!G0FG1 between the fundamental groups, for iD0; 1 ; see [8]. Also, p1L0˚1 2 consists of a disjoint union of copies of the universal cover z L0 of L0 , as the inclusion L0˚1 2,!X induces a monomorphism F ,!G0FG1 . Let J be a connected component of p1Wn iD1ei˚1 2 p1L0˚1 2 . Observe that J is a copy of the Cayley graph of F , and consider the finite 2 –dimensional CW– complex Y0D.J I/ [Jf 1 2gz L0˚1 2 . Thus, the universal cover z X can be regarded as a push-out obtained from Sg2Ggz K0 , Sg2Ggz K1 and Sg2GgY 0 (compare with the description given in [13], and see also [15, Section 3]) where: (a) Sg2Ggz K0 and Sg2Ggz K1 are unions of copies of the “vertex spaces” z K0 and z K1 , respectively, such that gz K0g0z K0 if and only if g1g02G0 (and are disjoint otherwise) and gz K1g0z K1 if and only if g1g02G1 (and are disjoint otherwise). (b) Sg2GgY 0 is a union of copies of the “edge space” Y0 such that the subcomplex corresponding to J fig inside gY 0 for iD0; 1 is glued to gz Ki via a lift Q fi:g WJfig!gz Ki of the map fi . Furthermore Q fi:g Q fi:g0 if g1g02Im 'iŠF for iD0; 1 (and their images are disjoint otherwise). Next, fix a copy gY 0 and consider each map Q fi;g WJfig ! gz Ki for iD0; 1 . Observe that this map is nullhomotopic (as Jfig is a finite connected 1 –dimensional complex and gz Ki is simply connected), so we can replace it (without altering the homotopy type of the entire construction) by a constant map hi;g WJ fig ! gz Ki whose image is a vertex inside Im Q fi;g . We do the same for any other copy g0Y0 and maps Q fi;g0 for iD0; 1 via homotopies which may be taken as translates (within z X ) of those for gY 0 and Q fi;g , iD0; 1 . Since the G –action on z X is properly discontinuous, the collection of all these homotopies together with the gluing lemma [1, Lemma I.4.9]yields a proper homotopy equivalence between z X and a new push-out y X in the proper category, where y X can be seen as the 2 –dimensional CW–complex obtained from a collection of copies of YD†J [Jf 1 2gz L0˚1 2 (here “ † ” stands for “suspension”) and the collection of copies gz K0 and gz K1 of z K0 and z K1 glued together appropriately through the suspension vertices of the copies of †J Y (via the image of the new maps hi;g ) Algebraic & Geometric Topology, Volume 23 (2023) 6Manuel Cárdenas, Francisco Fernández Lasheras, Antonio Quintero and Ranja Roy z K0z K1 p q Y Y Y Y Y Y Figure 1 in such a way that the copies of Y inside y X are in a bijective correspondence with the copies gY 0 in the construction of z X . Observe that the universal cover z X is modeled after the Bass–Serre tree for GDG0FG1 , but the group G is no longer acting on y X . Nevertheless, y X still keeps that same tree-like structure. See Figure 1 for a pictorial description of y Xin the infinite ended case. We choose the copy of Y in y X (which we again refer to as Y ) corresponding to the copy Y0 for z X (ie g1 ), and consider the copies z K0 and z K1 in y X of the universal cover of K0 and K1 , which intersect Y at vertices p2z K0 and q2z K1 , taken as basepoints. We consider filtrations by compact subsets Ci;1 Ci;2      z Ki with p2C0;1 and q2C1;1 , and such that no vertex of z Ki is in the boundary of any of the Ci;j . These Ci;j may be taken as finite subcomplexes in some barycentric subdivision of z Ki . We proceed to build a filtration by compact subsets C1C2     y X inductively. The subset C1 consists of Y[C0;1 [C1;1 . Assume Cn is constructed. Then, CnC1 is the union of Cn[C0;nC1[C1;nC1 , the copies of Y which intersect Cn , and the translates of C0;nC1 and C1;nC1 on all those copies gz K0 and gz K1 which may intersect these copies of Y at a vertex (so that the corresponding basepoint on z Ki is sent to the corresponding intersection vertex in gz Ki). Observe that the fundamental pro-group of GDG0FG1 at each end is represented by pro-1.z X/ Špro-1.y X/  ff1g 1.y XC1/ 1.y XC2/    g; where the basepoints are taken on any base ray determining the given end. We will show that if z K0 is noncompact then z K0 (and hence the group G0 ) is semistable at each end; the proof for G1 is analogous. In fact, one can define continuous retractions Algebraic & Geometric Topology, Volume 23 (2023) Proper 2–equivalences between infinite ended finitely presented groups 7 hjWy XCj!z K0C0;j as follows. Fix pj2z K0C0;j (along a given base ray). If z2y XCj is in z K0C0;j then we set hj.z/ Dz . Otherwise, define hj.z/ in the following way. Let  be a path in y X from z to p , and let w be the vertex at which  meets z K0 for the first time. If w…C0;j then we define hj.z/ Dw , otherwise, we set hj.z/ Dpj . One can check that each hj is a continuous retraction, by the choice of the compact subsets C0;j z K0 . Observe that hj is not a proper map, as it maps noncompact subsets of each copy of the universal cover of Ki (except for z K0 ) to a point; in fact, if a copy of Y is being attached to a vertex w of z K0C0;j , then the connected component of y X fwg whose closure contains that copy of Y is sent by hj to the single point w . However, as a loop in y X is a product (modulo change of basepoints) of loops each of which lives inside some copy of Y , gz K0 or gz K1 , one can check that these maps hjlead us to commutative diagrams 1.z K0C0;j /  1.z K0C0;j C1/ oo  1.y XCj/ .hj/  1.y XCjC1/ oo .hjC1/  1.z K0C0;j / 1.z K0C0;j C1/ oo where the unmarked arrows are induced by the inclusions, and the composition of any two consecutive vertical arrows is the corresponding identity homomorphism. Therefore the homomorphisms f.hj/gj1 are all surjective, and the conclusion of the first part of the proposition follows as G is semistable at each end and hence pro- 1.y X/ ( Špro-1.z X/ ) is pro-isomorphic to a tower whose bonding maps are also surjective homomorphisms; see Section 2. In fact, given the filtrations above, one can easily check that a repeated use of van Kampen’s theorem yields that each vertical inclusion-induced homomorphism (in the diagram above) is an isomorphism (and hence so is each .hj/ ), by construction of y X. In the case of an HNN-extension HFD hH; tIt1 0.ei/t D 1.ei/; 1 ini with monomorphisms iWF!H for iD0; 1 , let K be a finite 2 –dimensional CW–complex with 1.K/ ŠH and fiWWn iD1S1!K for iD0; 1 be cellular maps such that Im fi1.K/ corresponds to the subgroup Im iH . Let L be the 2 –dimensional CW–complex constructed as above and consider the adjunction space XDL[f0f0g[f1f1gK , with 1.X/ ŠHF . Then the proof is similar to the one given above for the amalgamated product. Algebraic & Geometric Topology, Volume 23 (2023) 8Manuel Cárdenas, Francisco Fernández Lasheras, Antonio Quintero and Ranja Roy For the second part of the proposition, observe that the fundamental pro-group of G at a certain end can also be thought of as the fundamental pro-group of y X at the corresponding end, as the universal cover z X is proper homotopy equivalent to y X . Moreover, if G is semistable at each end then so is the 2 –dimensional CW–complex y X . Consider a base ray !WŒ0; 1/!y X with !.tn/2y XCn for n1 . We are to study pro-1.y X; !/. For this, we will distinguish the following two cases: Case 1 There is a copy gz Kiy X for iD0; 1 such that for all t0 there exists t0t with !.t0/2gz Ki . In particular, the end determined by ! can be represented by a sequence of points within the copy gz Kiy X . In this case, we can consider a reparametrization !0 of ! such that !0.k/ 2gz Ki for all k2N (and yielding proisomorphic fundamental pro-groups). On the other hand, we may also consider a proper ray !00 WŒ0; 1/!gz Ki satisfying !00.k/ D!0.k/ for all k2N , and hence defining the same end of y X as !0 (and ! ). By the semistability hypothesis, we have a pro-isomorphism pro- 1.y X; !0/Špro-1.y X; !00/ ; in fact, it is not hard to show that two such proper rays !0 and !00 are always properly homotopic within y X ; compare with [10, Lemma 4]. Finally, taking similar filtrations as in the first part of the proof, one can easily check that a repeated use of van Kampen’s theorem yields a pro-isomorphism pro- 1.y X; !00/Špro-1.g z Ki; !00/ , by construction of y X . Thus, this relates the fundamental pro-group of G at a certain end (determined by ! ) to the fundamental pro-group of one of the factors Gi at one of its ends. In the case of an HNN-extension the argument is similar. Case 2 For every copy gz Kiy X of z Ki for iD0; 1 there exists ti;j 0 such that !.Œti;j ;1//\gz KiD∅ (ie ! eventually leaves any copy gz Ki inside y X ). In this case, by construction, one can easily check that for each n1 there exists .n/ n (sufficiently large) so that every loop in the component of y XC.n/ which contains !.t.n// is homotopic within the component of y XCn which contains !.tn/ to a constant map (with image one of the suspension vertices of some copy of Yy X ), and hence the inclusion-induced homomorphism 1.y XC.n/; !.t.n/// !1.y XCn; !.tn// is trivial. Thus, the fundamental pro-group pro- 1.y X; !/ , which is also represented by the tower (see [4, Section 16.2]) f1g 1.y XC1; !.t1// 1.y XC.1/; !.t.1/// 1.y XC2.1/; !.t2.1///    ; is pro-isomorphic to the trivial tower. Therefore, if the fundamental pro-group of G at a certain end is nontrivial, then Case 1 above must hold and the conclusion follows. Algebraic & Geometric Topology, Volume 23 (2023)