Quasi-isometries and proper homotopy: The quasi-isometry invariance of proper 3-realizability of groups

PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 147, Number 4, April 2019, Pages 1797–1804 https://doi.org/10.1090/proc/14373 Article electronically published on December 12, 2018 QUASI-ISOMETRIES AND PROPER HOMOTOPY: THE QUASI-ISOMETRY INVARIANCE OF PROPER 3-REALIZABILITY OF GROUPS M. C´ ARDENAS, F. F. LASHERAS, A. QUINTERO, AND R. ROY (Communicated by Ken Bromberg) Abstract. We recall that a finitely presented group Gis properly 3-realizable if for some finite 2-dimensional CW-complex Xwith π1(X)∼ =G, the universal cover  Xhas the proper homotopy type of a 3-manifold. This purely topological property is closely related to the asymptotic behavior of the group G.Weshow that proper 3-realizability is also a geometric property meaning that it is a quasi-isometry invariant for finitely presented groups. In fact, in this paper we prove that (after taking wedge with a single n-sphere) any two infinite quasiisometric groups of type Fn(n≥2) have universal covers whose n-skeleta are proper homotopy equivalent. Recall that a group Gis of type Fnif it admits aK(G, 1)-complex with finite n-skeleton. 1. Introduction Generally, we will be working within the category of locally finite CW-complexes and proper maps. 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. Given an infinite (strongly) locally finite CW-complex Y,aproper ray in Yis a proper map ω:[0,∞)−→ Y. We say that two proper rays ω,ωdefine the same end if their restriction to the natural numbers ω|N,ω|Nare properly homotopic. This equivalence relation gives rise to the notion of end determined by ωas the corresponding equivalence class, as well as the space of ends E(Y)ofYas a compact totally disconnected metrizable space (see [3,14]). The CW-complex Yis semistable at the end determined by ωif any other proper ray defining the same end is in fact properly homotopic to ω. The number of ends of a finitely generated group Grepresents the number of ends of the (strongly) locally finite CW-complex  X1for some (equivalently any) CW-complex Xwith π1(X)∼ =Gand with finite 1-skeleton. This number equals 0,1,2, or ∞(finite groups have 0 ends; see [14,21]). Here,  Xdenotes the universal cover of X. We say that a finitely presented group Gis semistable at each end (resp., at infinity,ifGis 1-ended) if the (strongly) locally finite CW-complex  X2is Received by the editors April 10, 2017, and, in revised form, July 10, 2018. 2010 Mathematics Subject Classification. Primary 57M07; Secondary 57M10, 57M20. Key words and phrases. Proper homotopy, quasi-isometry, properly 3-realizable, 3-manifold. This work was partially supported by the project MTM 2015-65397. c 2018 American Mathematical Society 1797 Licensed to Universidad de Sevilla. Prepared on Mon Dec 18 03:56:15 EST 2023 for download from IP 193.147.168.95. License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 1798 M. C ´ ARDENAS, F. F. LASHERAS, A. QUINTERO, AND R. ROY so, for some (equivalently any) CW-complex Xwith π1(X)∼ =Gand with finite 2- skeleton. Observe that any finite-dimensional locally finite CW-complex is strongly locally finite; see [14, Prop. 10.1.12]. We recall that a finitely presented group Gis properly 3-realizable if for some finite 2-dimensional CW-complex Xwith π1(X)∼ =G, the universal cover  Xof Xhas the proper homotopy type of a 3-manifold M(with boundary). Notice that it follows from [6] that  Xalways has the proper homotopy type of a 4-manifold. This concept was originally motivated by the fact that Lefschetz duality then yields the freeness of the second cohomology H2(G;ZG), as it is a direct summand of H2 c(  X;Z)∼ = H2 c(M;Z)∼ =H1(M,∂M;Z)∼ =  H0(∂M;Z) (see [15]). This property is a virtual property which does not depend on the choice of Xafter taking wedge with a single 2-sphere S2[1]. It is worth mentioning that direct products and ascending HNN- extensions of infinite finitely presented groups are properly 3-realizable [11,17]. In fact, any extension of an infinite finitely presented group by another infinite finitely presented group is properly 3-realizable [10]. One-relator groups are also properly 3-realizable [9]. Also, it has been proved that the class of properly 3-realizable groups is closed under amalgamated products over finite groups [8], and contains the class of all groups whose fundamental pro-group at each end is semistable and pro-(finitely generated free) [17, 18]. Moreover, it is conjectured in [13] that those two classes are the same (and examples of non-proper 3-realizable groups are given), and it has been proved so in [13] under the QSF (quasi-simply filtered) property, i.e., roughly speaking, that the corresponding universal cover admits an exhaustion that can be “approximated” by simply connected complexes (see [5]). To this respect, the stronger conjecture that all finitely presented groups are QSF has been posed in [20] (and related papers). One basic notion in the asymptotic geometry of discrete groups is that of quasiisometry. In general, given two pseudometric spaces (X,d)and(X,d ), a quasiisometry between them is a (not necessarily continuous) function f:X−→ Xfor which there exist constants λ,  > 0andC≥0 satisfying: (i) 1 λd(x, y)−≤d(f(x),f(y)) ≤λd(x, y)+for all x, y ∈X. (ii) For every x∈X,thereexistsx∈Xso that d(f(x),x )≤C. In the case of a finitely generated group G, consider the word metric on Gwith respect to some finite generating set S⊂G;namely,d(g,h)=l(h−1g), where l(θ) stands for the length of θ∈Gas a word in S∪S−1. For a (connected) CW-complex Y, consider the following natural pseudometric ρon Y,whereρ(x, y) is defined as the least integer kso that there is a path in Yjoining xto ywhich meets the interior of k+ 1 different cells. It is worth mentioning that if Gis a finitely generated group of type Fn(together with some, equivalently any, generating set S)andXis a K(G, 1)-complex with finite n-skeleton, then  Xnand Gare quasi-isometric with respect to the (pseudo)metrics considered above (see [14, §18.2]). Again, here  Xdenotes the universal cover of X. One can think of this as a nexus between the asymptotic geometry and the asymptotic topology of discrete groups, since many topological invariants of  Xtranslate into topological invariants of the group G. Licensed to Universidad de Sevilla. Prepared on Mon Dec 18 03:56:15 EST 2023 for download from IP 193.147.168.95. License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use QUASI-ISOMETRY INVARIANCE OF PROPER 3-REALIZABILITY 1799 In this context, our main result is the following. Theorem 1.1. Let Gand Hbe two infinite finitely generated quasi-isometric groups of type Fn(n≥2). Then, for any K(G, 1)-complex Xand any K(H,1)- complex Y with finite n-skeleta there exists a proper homotopy equivalence between the universal covers  Xn∨Snand  Yn∨Sn. And its corollaries are the following. Corollary 1.2. Let Gand Hbe two infinite finitely presented quasi-isometric groups. Then, for any finite 2-dimensional CW-complexes Xand Ywith π1(X)∼ = Gand π1(Y)∼ =Hthere exists a proper homotopy equivalence between the universal covers  X∨S2and  Y∨S2. This readily follows from the spacial case n= 2 of Theorem 1.1, as any finite 2-dimensional CW-complex X, with π1(X)∼ =G, can be enlarged to a K(G, 1)- complex Wwith W2=X(see [14, Prop. 7.1.5]). Corollary 1.3. Let Gand Hbe two infinite finitely presented quasi-isometric groups. Then, Gis properly 3-realizable if and only if His so. In particular, if N≤Gis a finite normal subgroup, then Gis properly 3-realizable if and only if G/N is so. Observe that finite groups are all properly 3-realizable (see [1]). Thus, proper 3-realizability becomes a quasi-isometry invariant for finitely presented groups. 2. Proof of the main result It was proved in [4] that the number of ends and the semistability are quasiisometry invariants for finitely presented groups. In fact, these results are consequences of the following proposition. Proposition 2.1 ([14, Thm. 18.2.11]).Let Gand Hbe two finitely generated groups of type Fn(n≥2),letXbe a K(G, 1)-complex with finite n-skeleton, and let Ybe aK(H,1)-complex Y with finite n-skeleton. If Gand Hare quasi-isometric, then the universal covers  Xnand  Ynare proper n-equivalent. Apropern-equivalence is used here in the sense of [14]. Namely, we have the following. Definition 2.2 ([14, §11.1]).A proper cellular map f:X−→ Ybetween finitedimensional locally finite CW-complexes is a proper n-equivalence if there is another proper cellular map g:Y−→ Xsuch that the restrictions g◦f|Xn−1and f◦g|Yn−1 are proper homotopic to the inclusion maps Xn−1⊆Xand Yn−1⊆Y. An n-equivalence as in Definition 2.2 is a stronger version of the notion of a proper (n−1)-type which extends the classical (n−1)-type in ordinary homotopy theory introduced by J. H. C. Whitehead [22]. More precisely, we have the following. Definition 2.3. Let Xand Ybe two finite-dimensional locally finite CW-com- plexes. We say that they have the same proper n-type if there exist proper maps f:Xn+1 −→ Yn+1 and g:Yn+1 −→ Xn+1 such that the restrictions g◦f|Xnand f◦g|Ynare proper homotopic to the inclusion maps Xn⊆Xn+1 and Yn⊆Yn+1. Licensed to Universidad de Sevilla. Prepared on Mon Dec 18 03:56:15 EST 2023 for download from IP 193.147.168.95. License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 1800 M. C ´ ARDENAS, F. F. LASHERAS, A. QUINTERO, AND R. ROY Obviously, two n-dimensional CW-complexes are properly n-equivalent if and only if they have the same proper (n−1)-type. Recall that, as an immediate consequence of the proper cellular approximation theorem (see [3, 14]), proper 2- equivalences preserve the spaces of ends. In fact, in what follows, all maps can be assumed to be cellular. We have the following lemma. Lemma 2.4. Let ϕ:X−→ Xand ψ:Y−→ Ybe proper homotopy equivalences between n-dimensional locally finite CW-complexes. Then, Xand Yare proper n-equivalent (or, equivalently, they have the same proper (n−1)-type)if and only if Xand Yare so. Indeed, if ϕand ψare cellular and f:X−→ Yis a proper n-equivalence, then so is ψ◦f◦ϕ:X−→ Y,whereϕis a (cellular) proper homotopy inverse of ϕ. Recall that we work within the category of (strongly) locally finite CW-complexes and proper maps. Although this category is not closed under push-outs, it contains enough ones to allow the basic homotopical constructions. Given a locally finite tree T,byaproperly T-based CW-complex we mean a pair (X,ω)whereω:T−→ Xis a proper map. Moreover, (X,ω)issaidtobeproperly well T-based if ω is a proper cofibration; i.e., ωhas the proper homotopy extension property (for instance, a cellular embedding). By using the mapping cylinder of ω,anyproperly T-based CW-complex (X, ω) is a proper deformation retract of a properly T-based CW-complex (X,ω) such that ω:T−→ Xis properly homotopic to a cellular embedding. Arelative proper map f:(X,ω)−→ (Y,ω) is a proper map between properly T-based CW-complexes such that f◦ω=ω. Similarly, a relative proper homotopy between relative proper maps f,g :(X,ω)−→ (Y,ω)isaproperhomotopyH: X×I−→ Ysuch that Ht:(X,ω)−→ (Y,ω) is a relative proper map for any t. In order to deal with all the ends of Xsimultaneously, we define a CW-complex under Tto be a properly well T-based CW-complex (X,ω) such that ωis endfaithful; i.e., ωinduces a homeomorphism ω∗:E(T)−→ E (X) between the spaces of ends. A map under Tis a relative proper map f:(X,ω)−→ (Y,ω) between CW-complexes under T. A relative homotopy between maps under Tis termed a homotopy under T. Given a locally finite tree T, relative n-types are defined similarly for properly T- based CW-complexes (X,ω)and(Y,ω) by requiring the homotopies to be relative to the proper maps ω:T−→ Xand ω:T−→ Y. We next show that proper n-equivalences between n-dimensional CW-complexes can be replaced by suitable relative n-equivalences. Let us start by observing that the argument in the proof given in [19, Chap. 6, §5] (see also [12, I.2.3]) of a theorem of Dold works for proper maps and proper homotopies and it also shows the following result. Proposition 2.5. Let f:X−→ Ybe a proper n-equivalence and let i:A−→ X and j:A−→ Ybe proper cofibrations with dim A≤n−1and f◦i=j.Then, f:(X,i)−→ (Y,j)is a relative proper n-equivalence. Indeed, as in [19, Chap. 6, §5], the proof of Proposition 2.5 reduces to the following. Given a (relative) proper map f:X−→ Xwith f◦i=iand f|Xn−1 proper homotopic to the inclusion Xn−1⊆X, it suffices to obtain a (relative) proper map g:X−→ Xwith g◦i=iandsuchthatg◦f|Xn−1is proper homotopic to Licensed to Universidad de Sevilla. Prepared on Mon Dec 18 03:56:15 EST 2023 for download from IP 193.147.168.95. License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use QUASI-ISOMETRY INVARIANCE OF PROPER 3-REALIZABILITY 1801 the inclusion Xn−1⊆Xrel. A. The rest of the proof goes as in [19, Chap. 6, §5], using the fact that the inclusion Xn−1⊆Xis a proper cofibration as well. Proposition 2.6. Let n≥2, and assume that the n-dimensional locally finite CW- complexes Xand Yare proper n-equivalent (or, equivalently, they have the same proper (n−1)-type). Then, for the inclusion i:T−→ Xof any end-faithful tree T⊂Xthere exists an n-dimensional locally finite CW-complex (Y,i )under T,in the proper homotopy class of Y, such that (X, i)and (Y,i )have the same relative (n−1)-type. Proof. Let i:T−→ Xbe the inclusion. Given a proper n-equivalence f:X−→ Y,letX=Miand Y=Mf◦ibe the corresponding mapping cylinders, and let  f:X−→ Ybe the proper map induced by f. Notice that dim X=dimY=n. Let i1:T−→ Xand i 1:T−→ Ydenote the cofibrations given by the cellular embeddings of Tas the top copies in the corresponding mapping cylinders. Notice that the canonical projection p:( X,i1)−→ (X,i) is a proper map under Twhich is a proper homotopy equivalence. Hence, by using Dold’s theorem [3, Thm. I.4.16] (also directly derived from [19, Chap. 6, §5]) one gets that pis in fact a proper homotopy equivalence under T.Letq:(X, i)→(X,i1) be a homotopy inverse of p under T. Then, the composite f=  f◦q:(X,i)−→ (Y,i  1) is a proper map under T, which is a proper n-equivalence by Lemma 2.4. Moreover, Proposition 2.5 yields that fis a relative proper n-equivalence (under T) between (X, i)and(Y,i ) with i=i 1. Recall that proper n-equivalences induce homeomorphisms between the spaces of ends.  The next proposition is the proper analogue of a theorem due to J. H. C. Whitehead (see [22, Thm. 14]). Proposition 2.7. Assume that the n-dimensional locally finite CW-complexes X and Yare proper n-equivalent, n≥2. Then, for any end-faithful tree T⊂Xthere exists an end-faithful proper map ω:T−→ Yand n-spherical T-objects Sn αand Sn β such that the wedges X∨Sn αand Y∨ωSn β, obtained via the inclusion i:T−→ X and ω, are proper homotopy equivalent. The proof of Proposition 2.7 requires some terminology from [3]. By an nspherical object Sn αunder Twe mean the space obtained from Tby attaching #α−1(v)n-spheres at each vertex v∈T0,whereα:E−→ T0is a proper map from a countable discrete space E. Similarly, by attaching copies of the n-ball Bnwe get the corresponding T-object Bn α. Any finite-dimensional locally finite CW-complex (X,i) under Tis proper homotopy equivalent under Tto a T-reduced and T-normalized CW-complex (X,i) in the sense of [3]; i.e., X1=S1 δ1is a 1- spherical T-object and Xnis the result of attaching a T-object Bn δnvia a proper map fn:Sn−1 δn−→ Xn−1. This follows from [3, Chap. IV, §5]. The classification of spherical objects in [3, Prop. II.4.5] yields a kind of ∞- element; namely, for any n-spherical T-object Sn αthere is relative proper homotopy equivalence Sn α∨Sn T≃Sn T. The spherical object Sn Tis defined by attaching exactly one n-sphere at each vertex v∈T0. Here, “∨” denotes the obvious wedge obtained by identifying the corresponding copies of Tin Sn αand Sn T. Proof of Proposition 2.7.By the observation above, we can assume that the ndimensional CW-complexes (X, i)and( Y,i ) under Tgiven by Proposition 2.6 are Licensed to Universidad de Sevilla. Prepared on Mon Dec 18 03:56:15 EST 2023 for download from IP 193.147.168.95. License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 1802 M. C ´ ARDENAS, F. F. LASHERAS, A. QUINTERO, AND R. ROY T-reduced and T-normalized. Then, by [23, Satz 6.11] we get n-spherical T-objects Sn αand Sn βand a relative proper homotopy equivalence under Tbetween the wedges X∨Sn αand Y∨Sn β. Thus, by the gluing lemma (see [3, Lemma I.4.9]) applied to a proper homotopy equivalence f:Y−→ Y, we get proper homotopy equivalences X∨Sn α≃Y∨Sn β≃Y∨ωSn β,whereω=f|T. Remark 2.8.The crucial ingredient for the proof of Proposition 2.7 is [23, Satz 6.11] whose proof is quite involved and makes use of homology with local coefficients in the proper setting. The argument of that proof has been adapted in [2, Thm. II.6.1] for an alternative proof of the classical Whitehead Theorem [22, Thm. 14]. An alternative and purely homotopic proof of Proposition 2.7 for the special case n= 2 (and hence of Corollary 1.2, as seen below) follows from [7, Prop. 6.3] whose proof works step by step for (X, i)and( Y,i ), with the tree Tplaying now the role of the half-line R≥0. The proof relies on the obvious co-H-space structure of the 1-skeleton of a T-normalized and T-reduced CW-complex, which is not the case for the (n−1)-skeleton when n≥3. We are now ready to prove Theorem 1.1. Proof of Theorem 1.1.Let Gand Hbe two infinite finitely generated quasi-iso- metric groups of type Fn(n≥2), and let Xbe a K(G, 1)-complex with finite nskeleton and Ybe a K(H, 1)-complex with finite n-skeleton. We may assume that the 0-skeleta of the CW-complexes Xand Yreduce to a single vertex, and let Tbe a maximal tree in the universal cover  Xnof Xn. By combining Propositions 2.1 and 2.7, we find n-spherical T-objects Sn αand Sn βand a proper homotopy equivalence  Xn∨Sn α≃  Yn∨ωSn β, for some end-faithful proper map ω:T−→  Yn.LetSn Tbe the ∞-element among the n-spherical T-objects. Then, we have  Xn∨Sn T≃  Xn∨(Sn α∨Sn T)≃  Yn∨ω(Sn β∨Sn T)≃  Yn∨ωSn T. We may assume that ωis a cellular map, so that ω(T0)⊂(T)0,whereT⊂  Y is a maximal tree in the universal cover of Yn. Then, it is clear that  Yn∨ωSn T coincides with  Yn∨Sn ω0,whereSn ω0is the n-spherical T-object determined by ω0=ω|T0, i.e., the one obtained by attaching #ω−1 0(v)n-spheresateachvertex v∈ω0(T0)⊂(T)0.LetT(ω0)⊂Tdenote the subtree generated by the vertices in the image of ω0. We will show below that the space of ends of T(ω0)coincides with the space of ends of T, and hence Sn ω0is proper homotopy equivalent under Tto the ∞-element Sn Tin the class of n-spherical T-objects, by the classification of spherical objects in [3, Prop. II.4.5]. Thus, by the gluing lemma ([3, Lemma I.4.9]),  Xn∨Sn T≃  Yn∨ωSn T=  Yn∨Sn ω0≃  Yn∨Sn T. Finally, we observe that  Xn∨Sn T= Xn∨Sn, and similarly  Yn∨Sn T= Yn∨Sn. It remains to check that the spaces of ends E(T(ω0)) of T(ω0)andE(T)ofT are the same. For this, let ω:T−→ Tbe the proper cellular extension of ω0 defined by setting, for any edge σ=(v,v), ω(σ) to be the single arc in Tfrom ω(v)toω(v). Observe that the image of ωis the subtree T(ω0) above. Moreover, Licensed to Universidad de Sevilla. Prepared on Mon Dec 18 03:56:15 EST 2023 for download from IP 193.147.168.95. License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use QUASI-ISOMETRY INVARIANCE OF PROPER 3-REALIZABILITY 1803 we have a commutative diagram E(T(ω0)) j∗//E(T) i∗  E(T) ω∗ 99 t t t t t t t t t ω∗// ω∗ OO E(Y) where iand jare the corresponding inclusions and ωis the restriction of ωonto its image. In particular, ω∗is surjective. The induced maps i∗and ω∗are homeomorphisms, as iand ωare end-faithful, and hence so is ω∗. It follows then that ω∗ is injective, and hence ω∗and j∗are also homeomorphisms.  Proof of Corollary 1.3.Let Xand Ybe finite 2-dimensional CW-complexes with π1(X)∼ =Gand π1(Y)∼ =H. By Corollary 1.2, there is a proper homotopy equivalence  X∨S2≃ Y∨S2. Assume that Gis a properly 3-realizable group. Then,  X∨S2≃ Y∨S2has the proper homotopy type of a 3-manifold, by [1, Prop. 1.3]. Since π1(Y∨S2)∼ =H, it follows that His a properly 3-realizable group as well. The second part follows immediately from a well-known corollary of ˇ Svarc- Milnor’s Lemma; see [14, Thm. 18.2.15] or [16].  References [1]R.Ayala,M.C´ardenas, F. F. Lasheras, and A. Quintero, Properly 3-realizable groups, Proc. Amer. Math. Soc. 133 (2005), no. 5, 1527–1535, DOI 10.1090/S0002-9939-04-07628-2. MR2111954 [2] Hans Joachim Baues, Combinatorial homotopy and 4-dimensional complexes,DeGruyter Expositions in Mathematics, vol. 2, Walter de Gruyter & Co., Berlin, 1991. With a preface by Ronald Brown. MR1096295 [3] Hans-Joachim Baues and Antonio Quintero, Infinite homotopy theory,K-Monographs in Mathematics, vol. 6, Kluwer Academic Publishers, Dordrecht, 2001. MR1848146 [4] Stephen G. Brick, Quasi-isometries and ends of groups, J. Pure Appl. Algebra 86 (1993), no. 1, 23–33, DOI 10.1016/0022-4049(93)90150-R. 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Tarfia s/n 41012-Sevilla, Spain Email address:[email protected] Departamento de Geometr´ ıa y Topolog´ ıa, Fac. Matem´ aticas, Universidad de Sevilla, C/. Tarfia s/n 41012-Sevilla, Spain Email address:[email protected] Departamento de Geometr´ ıa y Topolog´ ıa, Fac. Matem´ aticas, Universidad de Sevilla, C/. Tarfia s/n 41012-Sevilla, Spain Email address:[email protected] College of Arts and Sciences, New York Institute of Technology, Old Westbury, New York 11568-8000 Email address:[email protected] Licensed to Universidad de Sevilla. Prepared on Mon Dec 18 03:56:15 EST 2023 for download from IP 193.147.168.95. License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use