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On the Classifying Space for Proper Actions of Groups with Cyclic Torsion

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On the Classifying Space for Proper Actions of Groups with Cyclic Torsion On the classifying space for proper actions of groups with cyclic torsion Yago Antol´ınand Ram´on Flores September 17, 2018 Abstract In this paper we introduce a common framework for describing the topological part of the Baum-Connes conjecture for a wide class of groups. We compute the Bredon homology for groups with aspherical presentation, one-relator quotients of products of locally indicable groups, extensions of Zn by cyclic groups, and fuchsian groups. We take advantage of the torsion structure of these groups to use appropriate models of the universal space for proper actions which allow us, in turn, to extend some technology defined by Mislin in the case of one-relator groups. 2010 Mathematics Subject Classification. Primary: 55N91; Secondary: 20F05, 20J05. Key words. Bredon homology, classifying space for proper actions, aspherical presentations, Hempel groups, Baum-Connes conjecture 1 Introduction In [28], Mislin computed the Bredon homology of one-relator groups with coeffi- cients in the complex representation ring. These homology groups were defined arXiv:1107.0566v2 [math.KT] 24 Oct 2011 in the sixties by Bredon in the context of equivariant Homotopy Theory. Since the statement by Baum-Connes of their famous conjecture (see [5] for a thor- ough account and section 5 here for a quick review), there has been a growing interest in the computation of the Bredon homology groups, as they give, via a spectral sequence, a very close approximation to the topological part of the conjecture. Moreover, they are reasonably accessible from the point of view of the computations. Let G be a discrete group. To deal with the topological part of Baum-Connes conjecture, it is necessary to recall some basics of the theory of proper G-actions. A model for the classifying space for proper G-actions E G is a G-CW-complex X with the property that, for each subgroup H of G, the subcomplex of fixed points is contractible if H is finite, and empty if H is infinite. The latter condition means precisely that all cell stabilizers are finite, and, in this case, 1 we say that the G-action is proper. The model X for E G is important in our context because it is the target of the topological side of Baum-Connes conjecture. We also denote by (X)sing the singular part of X, that is, the subcomplex consisting of all points in X fixed by some non-trivial element of G, and we say that (X)sing is a model for (E G)sing. It is worth to note that both models for E G and (E G)sing are well-defined up to G-homotopy equivalence. In this paper we extend Mislin’s result [28, Corollary 3.23] to a wider class of groups. The key observation here is that the computation of the Bredon homology of one-relator groups does not use in full potential the shape of the concrete relation, but it relies only in two facts: the existence (up to conjuga- tion) of a unique maximal finite subgroup, and the construction of a model for E G whose singular part is zero-dimensional. We consider here the class of groups for which there is a finite family of Gcct cyclic subgroups such that every non-trivial torsion element of the group belongs to exactly one member of the family up to unique conjugation. If we consider the classical model X for E G as a bar construction for a certain G , it is easy ∈ Gcct to see (see Proposition 2.1) that for every non-trivial finite subgroup H < G the fixed-point set (X)H is just a vertex of the model. Hence the singular part of X is zero-dimensional, and we have all the needed assumptions to compute the Bredon homology groups (Proposition 4.2) and hence the Kasparov KK-groups (Proposition 5.2). Aside from the computations, the other main achievement of our article is the identification of well-known families of groups which belong to the class : Gcct groups with an aspherical presentation, one-relator products of locally indicable groups, some extensions of Zn for cyclic groups, and some fuchsian groups. For the first two families, moreover, we describe some particular models of E G which turn sharper our homology computations. For the particular class of Hempel groups (see Definition 3.1), we show that they have Cohen-Lyndon aspherical presentations, so we use Magnus induction and hierarchical decompositions to prove that Baum-Connes hold; then, our methods are also valid to compute top ∗ analytical K-groups Ki (Cr (G)) in this case. Note that the validity of the conjecture is not known for more general classes of groups of cohomological dimension two. The paper is structured as follows: in section 2 we formally introduce the class , as well as some background that will be needed in the rest of the Gcct paper; in section 3 we present some families of groups in the class, including some particular models of the classifying space for proper actions and inter- esting relationships with surface groups and other one-relator groups; section 4 is devoted to Bredon homology, which we describe in a little survey before undertaking our computations, and we finish in section 5 with the computation of the topological part of the Baum-Connes conjecture for these groups. Acknowledgments. Part of this paper is based on the Ph.D. thesis of 2 the first author at the Universitat Auton`oma de Barcelona. The first author is grateful to Warren Dicks for his help during that period. We are grateful to Ruben Sanchez-Garc´ıa for many useful comments and observations, and also to Brita E. A. Nucinkis, Ian Leary, Giovanni Gandini and David Singerman for helpful conversations. The authors were supported by MCI (Spain) through project MTM2008- 01550 and EPSRC through project EP/H032428/1 (first author) and project MTM2010-20692 (second author). 2 The class of groups Gcct In this article we will deal with groups that have, up to conjugation, a finite family of maximal malnormal cyclic subgroups. Precisely we deal with groups satisfying the following condition: (C) There is a finite family of non-trivial finite cyclic subgroups G such { λ}λ∈Λ that for each non-trivial torsion subgroup H of G, there exists a unique λ Λ and a unique coset gG G/G such that H 6 gG g−1. ∈ λ ∈ λ λ In particular, the groups Gλ are maximal malnormal in G. Recall that a subgroup H of a group G is malnormal if H gHg−1 = 1 for all g G H. ∩ { } ∈ − The class of groups G satisfying the condition (C) will be denoted by . Gcct Observe that any finite cyclic group and any torsion-free group is in . Gcct Moreover, this class is also closed by free products, so for example, the infi- nite dihedral group is in . In the following section we will describe many Gcct interesting examples. Our main objective is to describe the Bredon homology for a group G in . Our approach to this computation will be through the classifying space Gcct for proper G-actions, so we recall here a classical model which turns out to be very useful for our purposes. More sophisticated and particular models for E G will appear later in the article. 2.1 Proposition. Let G and G satisfying (C), then there exists a model { λ}λ∈Λ for E G with dim( sing) = 0. C C Proof. We adapt the proof of [20, Proposition 8] to our context. Let X be the left G-set gG : λ Λ, g G . A subgroup H of G fixes gG X if and only if { λ ∈ ∈ } λ ∈ g−1Hg G , so H is finite and, by condition (C), H fixes exactly one element ⊆ λ of X. Let E a model for the universal space EG (for example [14, Example 1B.7]). Recall that the join X E is the quotient space of X E [0, 1] under the ∗ × × identifications (x, e, 0) (x, e′, 0) and (x, e, 1) (x′, e, 1). The product X E ∼ ∼ × × [0, 1] is a G-set with G acting trivially in the interval [0, 1] and induces an action on X E. ∗ 3 Let H be a subgroup of G. If H is infinite, then H fixes no point of X or E. If H is non-trivial finite, it acts freely on X E (0, 1], and fixes exactly × × one point of X E 0 . Hence (X E)H is contractible and 0-dimensional. × × { } ∗ If H is trivial, then it fixes X E, which is contractible since E is contractible. ∗ Thus, = X E is a model for E G, such that dim( sing) = 0. C ∗ C 2.2 Remark. Our class of groups is a subclass of the groups with appro- Gcct priate maximal finite subgroups considered in [25, 4.11]. A particular model for E G is also provided there. A useful tool to show that a group G and a family of subgroups G of { λ}λ∈Λ G satisfy the condition (C) is the following theorem: 2.3 Theorem ([18, Theorem 6]). Let G be a group and G a finite family { λ}λ∈Λ of finite subgroups. If there exists an exact sequence of ZG-modules 0 Z[G/G ] P P P Z → λ ⊕ → n−1 →···→ 0 → Mλ∈Λ where P, Pn−1,...,P0 are ZG-projective, then for every finite subgroup H of G, there exists a unique λ Λ and a unique gGλ G/Gλ such that H is contained −1 ∈ ∈ in gGλg .
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