On the classifying space for proper actions of groups with cyclic torsion

arXiv:1107.0566v2 [math.KT] 24 Oct 2011 On the classifying space for proper actions of groups with cyclic torsion Yago Antol´ın and Ram´on Flores December 13, 2017 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 Znby 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 coefficients in the complex representation ring. These homology groups were defined 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 thorough 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 Gbe 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. Amodel for the classifying space for proper G-actions EGis a G-CW-complex Xwith the property that, for each subgroup Hof G, the subcomplex of fixed points is contractible if His finite, and empty if His 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 Xfor E Gis 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 Xfixed 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 Gand (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 conjugation) of a unique maximal finite subgroup, and the construction of a model for EGwhose singular part is zero-dimensional. We consider here the class Gcct of groups for which there is a finite family of 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 Xfor E Gas a bar construction for a certain G∈ Gcct, it is easy to see (see Proposition 2.1) that for every non-trivial finite subgroup H < G the fixed-point set (X)His just a vertex of the model. Hence the singular part of Xis 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 Znfor cyclic groups, and some fuchsian groups. For the first two families, moreover, we describe some particular models of E Gwhich 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 analytical K-groups Ktop i(C∗ r(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 Gcct, as well as some background that will be needed in the rest of the 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 interesting 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 {Gλ}λ∈Λsuch that for each non-trivial torsion subgroup Hof G, there exists a unique λ∈Λ and a unique coset gGλ∈G/Gλsuch that H⩽gGλg−1. In particular, the groups Gλare maximal malnormal in G. Recall that a subgroup Hof a group Gis malnormal if H∩gHg−1={1}for all g∈G−H. The class of groups Gsatisfying 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 infinite dihedral group is in Gcct.In the following section we will describe many interesting examples. Our main objective is to describe the Bredon homology for a group Gin Gcct. Our approach to this computation will be through the classifying space 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 Gand {Gλ}λ∈Λsatisfying (C), then there exists a model Cfor EGwith dim(Csing) = 0. Proof. We adapt the proof of [20, Proposition 8] to our context. Let Xbe the left G-set {gGλ:λ∈Λ, g ∈G}.A subgroup Hof Gfixes gGλ∈Xif and only if g−1Hg ⊆Gλ, so His finite and, by condition (C), Hfixes exactly one element of X. Let Ea model for the universal space EG (for example [14, Example 1B.7]). Recall that the join X∗Eis 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 Gacting trivially in the interval [0,1] and induces an action on X∗E. 3 Let Hbe a subgroup of G. If His infinite, then Hfixes no point of Xor E. If His non-trivial finite, it acts freely on X×E×(0,1],and fixes exactly one point of X×E×{0}.Hence (X∗E)His contractible and 0-dimensional. If His trivial, then it fixes X∗E, which is contractible since Eis contractible. Thus, C=X∗Eis a model for E G, such that dim(Csing) = 0. 2.2 Remark. Our class of groups Gcct is a subclass of the groups with appropriate maximal finite subgroups considered in [25, 4.11]. A particular model for EGis also provided there. A useful tool to show that a group Gand a family of subgroups {Gλ}λ∈Λof Gsatisfy the condition (C) is the following theorem: 2.3 Theorem ([18, Theorem 6]).Let Gbe a group and {Gλ}λ∈Λa finite family of finite subgroups. If there exists an exact sequence of ZG-modules 0→M λ∈Λ Z[G/Gλ]⊕P→Pn−1→ ··· → P0→Z where P, Pn−1,...,P0are ZG-projective, then for every finite subgroup Hof G, there exists a unique λ∈Λand a unique gGλ∈G/Gλsuch that His contained in gGλg−1. We remark that Alonso [1] find bounds for the dimension of the E Gof groups satisfying the hypothesis of the previous theorem. 3 Examples of groups in the class Gcct In this section we introduce some families of groups which will be proved to be in the class Gcct. Let us fix some notation first. Notation. We find useful to have different notations for a group given by a presentation and the presentation itself. We use a double bar hXkRito distinguish a presentation from the group being presented hX|Ri. Let Gbe a group and r∈G. If glies in a unique maximal infinite cyclic subgroup Cof G, we will denote by G √gthe unique generator of Cfor which g is a positive power. In this event, if gis the n-th power of G √g, we denote nby logG(g). If Xis a subset of G, we denote by GXthe image of Xunder left-conjugation by G, that is, GX={gxg−1:x∈X, g ∈G}.When X={x}we usually write Gxinstead of G{x}. 3.1 Groups with aspherical presentation Let Fbe a free group freely generated by a finite set X={x1,...,xn},let Rbe a subset of Fand G:=hX|Ri.In a free group, each element lies in a unique 4 maximal infinite cyclic subgroup of F. For r∈R, let Grbe the image of hF √ri in G, a finite cyclic subgroup of G. Recall that a CW-complex is aspherical if its universal covering is contractible. By the Hurewicz-Whitehead theorem a CW-complex is contractible if and only if it is acyclic and simply connected. There exist several concepts of aspherical presentations, see [10]. We will say that a presentation hXkRiis aspherical if the abelianized of FRis isomorphic to ⊕r∈RZ[G/Gr], and then a group is aspherical if it admits an aspherical presentation. It is a famous conjecture of Eilenberg-Ganea that the torsion-free aspherical groups are precisely the groups of cohomological dimension two. We now review the topological significance of asphericity. Recall that the Cayley graph of Gwith respect to Xis a G-graph Γ with vertex set Gand edge set G×X; for an edge e= (g, x) the initial vertex ιe is gand the terminal vertex τe is g·x. The augmented cellular chain complex of Γ is Z[G×X]∂ →ZG→Z→0 where ∂(g, x) = gx −g. It is well-known that the kernel of ∂is isomorphic to FRab and the kernel map θ:FRab →Z[G×X] is induced by the total free derivative ∂ ∂X :F→ Z[F×X],which is defined by f7→ (∂f ∂x1,..., ∂f ∂xn).The map f7→ ∂f ∂xiis a derivation from F→ZF, i.e. it satisfies the identity ∂f1f2 ∂xi=∂f1 ∂xi+f1∂f2 ∂xi.Hence ∂ ∂xiis uniquely determined by its values on X, and ∂xj ∂xiis equal to 0 if i6=jand 1 if i=j. See [9, Proposition 5.4] or [13, Corollary 9.4] for a proof. Hence, there is the following exact sequence of ZG-modules (1) 0 →FRab →Z[G×X]→ZG→Z→0. From now on we assume that hXkRiis aspherical, that is FRab ∼ = ⊕r∈RZ[G/Gr].This is the case, for example, when Rconsists of a single element by Lyndon’s identity theorem (see, [26]). Let R0⊂Rbe the set of r∈R for which Gr6= 1. Then, by Theorem 2.3, Gand {Gr}r∈R0satisfy the condition (C). With our assumption the sequence (1) becomes (2) 0 → ⊕r∈RZ[G/Gr]θ → ⊕x∈XZG∂ →ZG→Z→1. We describe now a model for E G, which is built exactly the same way as the usual for one-relator groups. The construction is basically the same as in [2, Review 7.4] which deal with a special case when |R|= 2, so we omit the details. 5 Recall that the Cayley complex of hXkRi,denoted C=ChXkRi,is a twodimensional CW-complex with exactly one 0-cell denoted [1], with set of 1-cells, denoted [X],in bijective correspondence with Xby a map denoted X→[X], x7→ [x],and with set of 2-cells, denoted [R], in bijective correspondence with Rby a map denoted R→[R], r 7→ [r].The attaching maps are determined by the 1-cells. Each ris a word in X±1,and we take the closure of the 2- cell [r] to be a polygon whose (counter-wise) boundary has the corresponding labeling in the 1-cells and their inverses, and this labeling gives the attaching map for [r]; the inverse of a 1-cell is the same 1-cell with the opposite orientation. The fundamental group of C,with base-point the unique 0-cell, has a natural identification with G=hX|Ri. Let Cbe the universal cover of C.The 1-skeleton of Cis the Cayley graph of Gwith respect to X. It can be checked that Cis simply connected and the augmented cellular complex of Cis the ZG-complex 0→Z[G×[R]]→Z[G×[X]] →Z[G×{[1]}]→Z→0. For [r]∈[R],let (g, [r]) be the lift of the 2-cell of [r] in Ccorresponding to the vertex g∈G. Let C′denote the CW-complex that is obtained from C by identifying the 2-cells (g, [r]) and (gF √r, [r]),for each g∈Gand r∈R. We denote this identified cell by (gGr,[r]).It can be checked that C′is again simply connected, and the augmented chain complex of C′is the exact sequence (2). Hence C′is contractible. If Gr={1}for each r∈R, that is R=F √R, then C=C′is acyclic and hence contractible; in particular Cis aspherical and Cis an E G. If F √R6=R, G does permute the open cells of C′, but C′is not a G-CW- complex since, for some r∈R, F √rfixes the 2-cell that is in the equivalence class of (1,[r]) but does not fix it pointwise since it does not fix the 1-cells where this cell is attached. Let C′′ denote the CW-complex obtained from C′by subdividing each 2-cell (gGr,[r]) into logFr2-cells. If we think (gGr,[r]) as a polygon with |r|-sides, the subdivision corresponds to adding a new vertex in the center, dividing from this vertex into logFrsubpolygons, such that F √rpermutes them. Notice that C′′ has one free G-orbit of 2-cells for each r∈R. Also, for each r∈R, we have also added a free G-orbit of 1-cells to C′.Finally for each r∈R we have added a G/Gr-orbit of 0-cells to C′.The CW-complex C′′ is then a contractible G-CW-complex whose augmented cellular chain complex is exact and has the form (3) 0 →ZG|R|→ZG|R|+|X|→ZG⊕(⊕r∈RZ[G/Gr]) →Z→0. Now by (C) for each non-trivial torsion subgroup Hof Gthere exists a unique r∈Rand gGr∈G/Grsuch that H≤gGr,and therefore Hfixes only the 0-cell obtained on the subdivision of the 2-cell (gGr,[r]) of C′.Then C′′ is an E G. 6 3.2 One-relator products of locally indicable groups Recall that a group is indicable if either is trivial or it has an infinite cyclic quotient. A group is locally indicable if every finitely generated subgroup is indicable. Notice that a locally indicable group is torsion-free. Let A, B be locally indicable groups having finite dimensional Eilenberg- MacLane spaces CAand CBrespectively. Let r∈A∗B, such that ris not conjugate to an element of Anor of B. It can be deduced, from Bass-Serre theory, that the centralizer CA∗B(r) is infinite cyclic, and hence we can define A∗B √r. Denote G= (A∗B)/A∗Brand Gr=hA∗B √ri/hri ≤ G. Let now CA∗Bbe the CW-complex obtained by attaching a 1-cell eto the disjoint union of CAand CB,where the endpoints are 0-cells in CAand CB; then CA∗Bis connected and has the homotopy type of an Eilenberg-MacLane space K(A∗B, 1).Choose a map φ:S1→ C(1) A∗Bthat represents A∗B √r, where C(1) A∗B denotes the 1-skeleton of CA∗B.We will assume that the basepoint for S1,goes under φto a vertex vof CA∗B. Let CGrbe a model for K(Gr,1).If Gr= 1 we may think of CGras a disk and p:S1→CGras the natural inclusion to the boundary. If Gris a non-trivial cyclic group of order logA∗B(r),CGris a CW-complex with one cell in each dimension and p:S1→CGris the natural projection. In [19, Theorem 1], it is showed that the following push-out (Figure 3.0.1) has the homotopy type of K(G, 1). In the sequel, the push-out of this diagram S1φ// p  CA∗B CGr Figure 3.0.1: The push-out that gives a K(G, 1). will be denoted CG. Using this construction, in [19, Proposition 7] Howie shows that there is a sequence of ZG-modules 0→Z[G/Gr]⊕P→Pn−1→ ··· → P0→Z where P, Pn−1,...,P0are ZG-projective, and thus the subgroup Grand Gsatisfy the condition (C) by Theorem 2.3. We now describe how to obtain an E Gfrom CGwith dim E Gsing ≤0.Let CGdenote the universal cover of CG.Observe that if A∗B √r=r, then Gis torsion-free, and we take CGas our model for E G. Assume then A∗B √r6=r, so we will modify CGto have all the singular action in dimension 0. Let αbe the 2-cell of CGrand let αdenote a lift of αin CG. 7 For n > 2 we remove G-equivariantly the orbit of the n-dimensional cells of CGcorresponding to the n-dimensional cell of CGr,and we identify two 2-cells gα and g′αif g′∈gGr.Notice that this two cells have the same boundary. We denote the 2-cell identified with αby Grα, and we claim that the obtained space, denoted by C′ G,is still contractible. The latter construction is not a G-CW-complex, because A∗B √rfixes the 2-cell Grαbut does not fix it pointwise since it does not fix the 1-cells where it is attached. So, to obtain a G-CW -complex, we subdivide each 2-cell gGrα like in the case of aspherical presentations. Formally, we remove the G-orbit of Grαand we add a G/Gr-orbit of 0-cells {gGru:g∈G}, and a G-orbit of 1-cells from {gf:g∈G}with fattached to gvto gGruand finally a Gorbit of 2-cells {gβ :g∈G}with gβ attached to the path that starts at gA∗B √rv,then goes through the edge gA∗B √rf,then goes through gf−1and finally through the subpath of the lift of φthat goes from gvto gA∗B √rv. We denote this new complex by C′′ G.Clearly it is a G-CW-complex. Let H be a finite subgroup of G, by (C) there exists a unique gGr∈G/Grsuch that H≤gGrg−1,and therefore Hfixes only the 0-cell gu. Then C′′ Gis an E Gand dim(C′′ G)sing = 0. Let us now prove the previous claim. We have to show that the space C′Gis simply connected and acyclic. Since the removed cells where attached to each other, the space remains simply connected. By construction of the push-out, we have a Mayer-Vietoris exact sequence which locally looks like ··· → H2(CA∗B)⊕H2(CGr)→H2(CG)→H1S1→H1(CA∗B)⊕H1(CGr)→... Since φis injective at the π1level, the map H1S1→H1(CA∗B)⊕H1(CGr) is inyective. Moreover, by construction of C′ G,for i≥2,Hi(C′G) = Hi(CA∗B). Hence for i≥2,Hi(C′G) = Hi(CA∗B) = Hi(CG) = 0.Since the boundary of α and A∗B √rα in the 1-skeleton is the same, the identification do not change the H1(CG).Then the space C′ Gis acyclic. Let (CA, dA) and (CB, dB) denote the integral cellular chain complexes of CAand CB,the universal covers of CAand CB,Then the integral cellular chain complex (C, d) of C′′Ghas the form at the module level •Ci∼ =(ZG⊗ZACA i)⊕(ZG⊗ZBCB i) for i≥2, •C1∼ =(ZG⊗ZACA 1)⊕(ZG⊗ZBCB 1)⊕Z[G×{e, f}], •C0∼ =(ZG⊗ZACA 0)⊕(ZG⊗ZBCB 0)⊕ZG/Gr. and dk:Ck→Ck−1are induced by (1 ⊗dA k)⊕(1 ⊗dB k) for k≥2. 3.3 Hempel Groups and Cohen-Lyndon asphericity In this subsection we discuss a family of two-relator groups wich include onerelator quotients of fundamental groups of surfaces. Originally, Hempel [15] and 8 Howie [16] studied one-relator quotients of the fundamental groups of orientable surfaces from a topological point of view. Hempel showed that such a group is an HNN -extension of a one relator group. These groups have recently attracted the attention of various authors, see [2], [7] or [17]. In [2] a family of two-relator groups with a presentation of the form (4) hx, y, z1,...,zdk[x, y]·u, ri, where d≥1, u ∈ hz1,...,zd| i and r∈ hx, y, z1,...,zd| i,was studied. By the classification of closed surfaces, any closed surface Swith Euler characteristic ≤ −2 can be decomposed as a connected sum of a torus and projective planes or a torus and more tori. In any case, the fundamental group of Sadmits a presentation of the form hx, y, z1,...,zdk[x, y]·uiwith u∈ hz1,...,zd| i. We will restrict to a subfamily of these two-relator groups, which we called Hempel groups. The definition involves some technicality, but one may think a Hempel group as a group admitting a presentation as (4) where rcannot be replaced by a positive power x. If Yis a subset of a set X, we say that g∈ hX| i involves some element of Yif there is some element of Yin the freely reduced word in Xrepresenting g. 3.1 Definition. Let d≥1, F =hx, y, z1,...,zd| i,and for f∈F, i ∈Z we denote yifby if. Let also X1:={1x} ∪ {izj:j= 1,...,d;i= 0,1,...}, u∈ hz1,...,zd| i and r∈F. We say that ris a Hempel relator for the presentation hx, y, z1,...,zdk[x, y]·ui,and that hx, y, z1,...,zdk[x, y]u, riis aHempel presentation, if the following conditions hold: (H1). The element rbelongs to hX1| i ⩽F. (H2). In hX1| i, r is not conjugate to any element of (0u)−1·(1x). (H3). With respect to X1, r is cyclically reduced. (H4). With respect to X1, r involves some element of {0z1,...,0zd}. A group Gadmitting a Hempel presentation will be called a Hempel group. 3.2 Lemma ([2, Lemma 5.4]).Let d≥1, F =hx, y, z1,...,zd| i, r ∈F, and u∈ hz1,...,zd| i.Then there exists w∈F, v ∈F([x, y]u)and α∈Aut(F) that fixes [x, y]and z1,...zdsuch that r′=w(vα(r)) is either a non-negative power of xor a Hempel relator for hx, y, z1,...,zdk[x, y]ui. 3.3 Example. Let Sbe the fundamental group of an orientable, compact, connected, closed surface of genus g≥2, and san element of S. Then S/ Ss is a Hempel group. 9 By (9) and (10) all the subgroups in F(λ) are, up to conjugation, cyclic subgroups of the cyclic Gλ,and the morphisms are given by restriction of representations, we have HF 0(G;RC) = Y λ∈Λ RC(Gλ). In the case Λ is empty, we have that HF 0(G;RC) = RC({1}) = Z. Now we deal with the higher homology groups. As seen before, a classifying space for proper actions E Gcan be used to compute HF ∗(G;RC).For example, it is known that the virtually free groups are exactly the groups which admit a tree as a model of E G, and their Bredon homology with coefficients in the complex representation ring has been described by Mislin in [28, Theorem 3.17]. We denote by BGthe orbit space (E G)/G, sometimes called “classifying space for G-proper bundles” (see [5, Appendix 3] for the motivation of the name). Note that an n-dimensional model for E Gproduces n-dimensional models for its orbit space. Moreover, we define dim(E G)sing to be the minimum of dim(Xsing) where Xis a model for E G. The following result relates the Bredon homology of E Gand the ordinary homology of BG: 4.1 Lemma ([28, Lemma 3.21]).Let Gbe an arbitrary group. Then there is a natural map HFin i(E G;RC)→Hi(BG;Z) which is an isomorphism in dimension i > dim(E G)sing + 1 and injective in dimension i= dim(E G)sing + 1. Now we are ready to state the main result of this section. 4.2 Theorem. Let Gbe a group on the class Gcct and let {Gλ}λ∈Λbe the subgroups for which condition (C) holds. Then (i). HF i(G;RC) = Hi(BG;Z)for i≥2. (ii). HF 1(G;RC) = (G/ Tor(G))ab where Tor(G)denotes the subgroup of Ggenerated by the torsion elements. (iii). HF 0(G;RC) = Qλ∈ΛRC(Gλ),if Λ6=∅or HFin 0(G;RC) = Z,if Λ = ∅. Proof. (i) follows from Lemma 4.1, since by Proposition 2.1 dim(E G)sing = 0. (iii) has been proved at the beginning of this section, so it only remains to prove (ii). By Proposition 2.1, we have a model Xof E Gsuch that dim(E G)sing = 0. Moreover, the vertex set of Xis in bijection with G⊔ {gGλ:λ∈Λ, g ∈G}. Hence, we have the following exact augmented cellular chain complex of ZG- modules ··· → ⊕J2ZG→ ⊕J1ZG→ZG⊕(⊕λ∈ΛZ[G/Gλ]) →Z→0 16 where Jiis the number of free G-orbits of i-dimensional cells in X. Also using the cellular structure of X, we have the following projective resolution of Zin ModFin-G ··· → C2(X)→C1(X)→C0(X)→Z→0 where Ci(X): DFin(G)→Ab, Ci(X)(G/H) = (0 if H6={1} ZGJiif H= 1 for i= 1,2,.... For i= 0 we have, C0(X): DFin(G)→Ab, C0(X)(G/H) = Z[G/Gλ] if H6= 1 where Gλis the unique subgroup that contains a conjugate of H, and C0(X)(G/{1}) = ZG⊕(⊕λ∈ΛZ[G/Gλ]). Let us write Cifor Ci(X)⊗FRCand C∗for the ordinary cellular chain complex C∗(X/G).Then we have: C0∼ =Z⊕(⊕λ∈ΛRC(Gλ)) , C0∼ =Z⊕(⊕λ∈ΛZ). We have the diagram in Figure 4.2, ker(π) mono  ∼ =//ker(ǫ) = Qλ∈Λ˜ RC(Gλ) mono  C2 d2// =  C1 d1// =  C0 epi // π  Qλ∈ΛRC(Gλ) ǫ  C2 d2//C1 d1//C0 epi //Qλ∈ΛZ Figure 4.2.1: Comparing the two chain complexes where π= (Id, ǫ) and ǫ:QRC(Gλ)→QZis the augmentation. Now we are ready to check the statements of the theorem. By [23, Proposition 3] (see also [4]), π1(BG) = G/ Tor(G) . Hence, H1(BG;Z)∼ = (G/ Tor(G))ab. By Lemma 4.1, the map HF 1(E G;RC)→H1(BG;Z) is injective. We have to show that it is surjective too. The diagram shows that ker d1= ker d1,thus ker d1/Im d2→ker d1/Im d2 is onto. 4.3 Remark. The model for E Gof [25, 4.11] is given by a G-pushout which involve EG and EGλfor λ∈Λ, so a Mayer-Vietoris argument may give some extra information about the higher homology groups Hi(BG;Z) appearing in Theorem 4.2 (i). 17 For the concrete families described above, we may give sharper statements: 4.4 Corollary. Let Aand Bbe locally indicable groups and r∈A∗Bnot conjugate to an element of Anor of B. Let G:= (A∗B)/A∗Brand let Grbe the cyclic subgroup of Ggenerated by the image of A∗B √r. Then (i). HF i(G;RC) = Hi(A;Z)⊕Hi(B;Z)for i > 2. (ii). HF 2(G;RC) = H2(BG;Z). (iii). HF 1(G;RC) = (G/ Tor(G))ab where Tor(G)denotes the subgroup of Ggenerated by the torsion elements. (iv). HF 0(G;RC) = RC(Gr)if Gr6={1}or HFin 0(G;RC) = Zif Gr={1}. Proof. By the discussion in the subsection 3.2, Gand Grare under the hypotheses of Theorem 4.2, and (ii),(iii) and (iv) follow directly. Statement (i) follows from the cellular chain complex of C′′ Gdescribed in 3.2. 4.5 Corollary. Let Fbe a free group and Rbe subset of F. Let G:=F/ FR and for each r∈R, let Grbe the cyclic subgroup of Ggenerated by the image of F √r. Suppose that FRab ∼ =⊕r∈RZ[G/Gr], and let T(R) = {r∈R:Gr6= 1}. Then (i). HF i(G;RC) = 0 for i > 2. (ii). HF 2(G;RC) = H2(G;Z). (iii). HF 1(G;RC) = (G/ Tor(G))ab where Tor(G)denotes the subgroup of Ggenerated by the torsion elements. (iv). HF 0(G;RC) = Qr∈T(R)RC(Gr)if T(R)6=∅or HFin 0(G;RC) = Zif T(R) = ∅. Proof. By the discussion in the subsection 3.1, Gand {Gr}r∈T(R)are again under the hypothesis of Theorem 4.2 and we have a 2-dimensional model for EGso (i),(iii) and (iv) follow. It only remains to prove the statement (ii). From the theorem we know that HF 2(G;RC) = H2(BG;Z),which, by the cellular chain complex (3) of C′′ described in 3.1, is isomorphic to the kernel of the induced map (ZG|R|→ZG|R|+|X|)⊗Z[G]Z. The complex C′described in 3.1 has cellular chain complex (2) and is homotopic to C′′,hence H2(BG;Z) is isomorphic to the kernel of the induced map (⊕r∈RZ[G/Gr]→ZG|X|)⊗Z[G]Z,which is isomorphic to H2(G, Z). In the case of Hempel groups, we can be even more precise: 18 4.6 Theorem. Let Gbe a Hempel group with presentation hx1,...,xkkw, ri, with k≥3, w ∈[x1, x2]hx3,...,xki ⊆ hx1,...,xk| i =Fand such that ris a Hempel relator for hx1,...,xkkwi.Let Gr:=CF(r)/hri=hF √r|ri.Then (i). HFin i(G;RC) = 0 for i > 2. (ii). HFin 2(G;RC) = H2(G;Z) = (Fr∪Fw∩[F, F ])/[F, Fr∪Fw] (iii). HFin 1(G;RC) = (hx1,...,xk|w, F √ri)ab. (iv). HFin 0(G;RC) = RC(Gr)if Gr6={1}and Zin the other case. Proof. By Theorem 3.10 we are in the hypothesis of Corollary 4.2. The last equality of (ii) is the classical Hopf identity. 5 Relation with Baum-Connes conjecture Let Hbe an aspherical group. Beyond their intrinsic interest, the results obtained in the previous section show their relevance in the context of Baum- Connes conjecture. More concretely, Corollary 4.5 will identify the equivariant version KH i(E H) of the K-homology of E H, as defined by Davis-L¨uck in [12]. Given an arbitrary countable discrete group G, the K-groups KG i(E G), which are defined via the non-connective topological K-theory spectrum, can be in turn identified with the Kasparov KK-groups KKG i(E G), which are constructed as homotopy classes of G-equivariant elliptic operators over E G. These homotopical invariants are related with the topological algebraic K- groups Ktop i(C∗ r(G)) of the reduced C∗-algebra of G, an object which is defined as the closure of a subalgebra of the Banach algebra B(l2(G)) of bounded operators over the space of square-summable complex functions over the group G, and whose nature is thus essentially analytic. Bott periodicity holds for both the homotopical and the analytical groups, and the relationship is given, for i= 0,1, by an index map: Θ : KKG i(E G)→Ktop i(C∗ r(G)), The Baum-Connes conjecture (or BCC for short) asserts that this index map is an isomorphism for every second countable locally compact group G. Originally stated in its definitive shape by Baum-Connes-Higson in [5], its importance come mainly from two sources: first, it relates two objects of very different nature, being the analytical one particularly inaccessible; and moreover, it implies a number of famous conjectures, as for example Novikov conjecture on the higher signatures, or the weak version of Hyman Bass’ conjecture about the Hattori-Stallings trace; see [28, Section 7] for a good survey on this topic. The conjecture BCC has been verified for an important number of groups, and 19 in particular for the groups in the class LHTH, which is defined by means of an analytical property (see [28, Section 5] for a detailed exposition). The class LHTH contains for example the soluble groups, finite groups and free groups; and as it also contains one-relator groups and it is closed under passing to subgroups and HNN -extensions, Theorem 3.10 implies that Hempel groups are in LHTH. On the other hand, every one-relator group is also aspherical (see Theorem 3.6 above). However, it is unknown if BCC holds for the class of aspherical groups, so it is interesting to investigate the value of the K-groups in both sides of the conjecture for aspherical groups. The main goal of this paragraph is to show how the results in the previous section allow to compute the topological side of the conjecture. The key result here, owed to Mislin [28, Theorem 5.27], is in fact a collapsed version of an appropriate Atiyah-Hirzebruch spectral sequence: 5.1 Theorem. Let Gbe an arbitrary group such that dim EG≤2. Then there is a natural short exact sequence: 0→RC(G)→KG 0(E G)→HFin 2(G;RC), and a natural isomorphism HFin 1(G;RC)≃KG 1(E G). Now the description of the Kasparov aspherical groups comes straight from Corollary 4.5, and generalises Corollary 5.28 from [28]: 5.2 Proposition. Let Gbe an aspherical group. Then KG 0(E G)fits in a short exact sequence RC(G)→KG 0(E G)→H2(G;Z) that splits, and moreover there is a natural isomorphism (G/ Tor(G))ab ≃KG 1(E G). In particular, BCC holds for Hempel groups, so we have also computed the analytical of the conjecture for these groups: 5.3 Proposition. Let Gbe a Hempel group with presentation hx1,...,xkkw, ri, with k≥3, w ∈[x1, x2]hx3,...,xki ⊆ hx1,...,xk| i =Fand such that ris a Hempel relator for hx1,...,xkkwi.Let Gr:=CF(r)/hri=hF √r|ri.Then KG i(E G)≃Ktop i(C∗ r(G)) for i= 0,1, there is a split short exact sequence RC(G)→KG 0(E G)→(Fr∪Fw∩[F, F ])/[F, Fr∪Fw] and a natural isomorphism (hx1,...,xk|w, F √ri)ab ≃KG 1(E G). Proof. It is a consequence of Theorem 4.6. 20 References [1] Juan M. Alonso, Quasi-projective and relative cohomological dimension of groups, Trans. Amer. Math. Soc. 325 (1991), no. 2, 715–739. [2] Yago Antol´ın, Warren Dicks and Peter A. Linnell, Non-orientable surfaceplus-one-relation groups, J. Algebra 326 (2011), 433. 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Yago Antol´ ın, School of Mathematics, University of Southampton, University Road, Southampton SO17 1BJ, UK E-mail address:[email protected] Ram´ on Flores, Departamento de Estad´ ıstica, Universidad Carlos III, Avda. de la Universidad Carlos III, 22 28270 Colmenarejo (Madrid), Spain E-mail address:[email protected] 23