ON CLASSIFYING SPACES FOR THE FAMILY OF VIRTUALLY CYCLIC SUBGROUPS

DANIEL JUAN-PINEDA AND IAN J. LEARY

Abstract. We construct a model for the universal space for Gactions with virtually cyclic stabilizers for groups G in a class that includes all word hyperbolic groups. We introduce the notation \underline{\underline{E}}G for this universal space.

1. introduction

Let G be a discrete group and let F be a family of subgroups of G, by which we mean a collection of subgroups of G closed under taking subgroups and conjugation by elements of G. A classifying space for F (or an F-classifying space, or a universal G-space with stabilizers in F) is a contractible space E(G, F) endowed with an action of G which satisfies that the fixed points E(G, F)^{H} is non empty and contractible if and only if H ∈ F. The importance of studying F-classifying spaces has been recorded in many instances, most notably in the cases when the family F consists of just the trivial subgroup of G, and when the family consists of all finite subgroups of G. In the case when F contains just the trivial subgroup of G, E(G, F) is just the total space of the universal principal G-bundle. In the case when F consists of all finite subgroups of G, then E(G, F) is the universal proper G-space. For any G and F, the space E(G, F) is a terminal object in a certain homotopy category of G-spaces with stabilizers in F. Another way in which these spaces arise is in the various formulations of assembly maps and assembly isomorphism conjectures [12, 14].

It can be shown that these classifying spaces exist [19] and that they are unique up to G-homotopy. However, specific models for E(G, F) with geometric properties suitable for applications are difficult to find for general families, although much work has been done for the family of finite subgroups[10, 17, 18, 20].

2000 Mathematics Subject Classification 19A31, 19B28, 19D35, 20F67 .

Key words and phrases: hyperbolic groups, K-theory.

Research partially supported by CONACyT and DGAPA research grants.

We are interested in this work in F classifying spaces in the case when F consists of the family of virtually cyclic subgroups of G. Recall that a group is called virtually cyclic if it is either finite or has an infinite cyclic subgroup of finite index.

Given a group G we will write \underline{\underline{E}}G for a model of a classifying space for the family of virtually cyclic subgroups of G. There are few examples of \underline{\underline{E}}G models which are suitable for computations see [6, 7]. Our main theorem in this work consists of constructing a specific model for \underline{\underline{E}}G for G in a class of groups which includes all groups that are hyperbolic in the sense of Gromov, or word hyperbolic. Using this model, we are able to compute the generalized homology theory associated to \underline{\underline{E}}G with coefficients in the K-theory functors 𝕂_{R} defined by Davis-Lück in [12]. Roughly speaking, the result says that for groups in this class, the generalized homology of \underline{\underline{E}}G is determined by that of \underline{E}G together with information about various Nil groups associated to the group rings of finite subgroups of G.

This result is of interest because according to the Farrell-Jones conjecture, it leads to a computation of the algebraic K-groups of the group algebra RG in terms of the same data. The Farrell-Jones conjecture has been verified for many rings R and groups G, [2, 6, 14, 20] including some of the groups that we consider.

The starting point for this paper was a model for \underline{\underline{E}}G appearing in [2] in the case when G is the fundamental group of a closed negatively curved manifold. Our construction can be viewed as a generalization of the construction given there. A different account of some of the results in our paper appears in Wolfgang Lück’s survey article on classifying spaces for families [21], but we include more examples and open questions, and we hope that the reader will find both accounts useful. The authors are grateful to Brian Bowditch for conversations concerning hyperbolic groups and to Holger Reich for discussions concerning [2].

2. Models for Universal Spaces

Let G be a discrete group, and let F denote a family of subgroups of G, by which we mean a collection of subgroups of G that is closed under conjugation by elements of G and under taking finite intersections.

Definition 1. A model for E(G, F) is a G-CW-complex E such that all cell stabilizers lie in the family F, and for any H ∈ F, the fixed point set E^{H} is contractible.

Remark 2. Often one adds the hypothesis that the family F be closed under taking subgroups. For F satisfying this hypothesis, the above definition is equivalent to: for all H ≤ G, E^{H} is contractible if H ∈ F and is empty if H ∉ F.

It may be shown that whenever X is any G-CW-complex with stabilizers in F and E is a model for E(G, F), there exists precisely one equivariant homotopy class of maps from X to E. In other words, a model for E(G, F) is a terminal object in the homotopy category of G-CW-complexes with stabilizers in F. In particular, any two models for E(G, F) are equivariantly homotopy equivalent. For this reason, E(G, F) is referred to as the universal G-space with stabilizers in F, and we will sometimes write ‘an E(G, F)’ instead of ‘a model for E(G, F)’.

One may show that there exist models for E(G, F) using a generalization of either Milnor’s or Segal’s construction of EG [23, 24]. For any set X, let E_{X} denote the infinite join E_{X} = X ∗ X ∗ X ∗ ⋯ of copies of X, and let E′_{X} denote the either the simplicial set or the semisimplicial set whose set of n-simplices is the set X^{n+1}, with face maps given by omitting a coordinate. (In the case of the simplicial set, the degeneracies are given by repeating a coordinate.) Each of E_{X} and E′_{X} is contractible if X is non-empty and is empty if X is empty. Now let X be a G-set such that each point stabilizer is in F, and such that for every H ∈ F, X^{H} ≠ ∅. Since (E_{X})^{H} = E_{X^{H}} and similarly for E′, it follows that each of E_{X} and E′_{X} is a model for E(G, F). (See [17] for more details.)

Example 1. When F contains just the trivial group, E(G, F) is the universal free G-CW-complex, usually called EG. Equivalently, EG is the universal cover of an Eilenberg-Mac Lane space BG = K(G, 1) for G.

Example 2. When F consists of all the finite subgroups of G, E(G, F) is often denoted by \underline{E}G. This G-space appears in the formulation of the Baum-Connes conjecture.

The following definition is motivated by the examples above.

Definition 3. Let \underline{\underline{E}}G denote E(G, F) in the case when F is the family of all virtually cyclic subgroups of G.

Just as \underline{E}G appears in the statement of the Baum-Connes conjecture [5], \underline{\underline{E}}G appears in the statement of other assembly conjectures such as the Farrell-Jones isomorphism conjectures [14]. For this reason, we feel that it is natural to introduce the notation \underline{\underline{E}}G.

Before describing some examples of \underline{\underline{E}}G, we pause to give a model for \underline{E}G for G virtually cyclic. The following proposition is fairly wellknown, but we include it for completeness and because we shall need to refer to it below.

Proposition 4. Let G be a virtually cyclic group. Then G contains a unique maximal normal finite subgroup N and exactly one of the following holds:

(1) the finite case, when N = G;

(2) the orientable case, when G/N is infinite cyclic;

(3) the non-orientable case, when G/N is infinite dihedral. In cases (2) and (3), there is a model for \underline{E}G with finitely many orbits of cells which is homeomorphic to the real line.

Proof. If G is not finite, let H′ be an infinite cyclic subgroup of G. This subgroup must have finite index, so let H be the intersection of the finitely many conjugates of H′. This H is a finite-index cyclic normal subgroup of G. Let K be the centralizer C_{G}(H) of H in G. Either K = G, or |G : K| = 2, since G/K embeds in Aut(H) which has order 2. Let E be a copy of the real line, with H acting via translation, so that a generator h for H acts as translation by 1. This action extends uniquely to an action of K: for any k ∈ K, pick the least n such that k^{n} = h^{m} for some m, and then k acts as translation by m/n. Let N be the kernel of the action of K on E. This N is finite (since N ∩ H = {0}), and since K/N acts faithfully by translations on E, it follows that K/N is infinite cyclic. Hence N consists of all of the elements of K of finite order, and so N is characteristic in K and normal in G. The only possibilities are that G/N = K/N is infinite cyclic, or G/N is infinite dihedral. In each case, the action of K on E extends to an action of G, and E may be made in to a G-CW-complex with one orbit of 1-cells and either one (in the orientable case) or two (in the non-orientable case) orbits of 0-cells. In the orientable case, N consists of all elements of G of finite order. In the non-orientable case, G contains infinitely many maximal finite subgroups, in two conjugacy classes, and each maximal finite subgroup contains N as an index two subgroup. □

3. Models for \underline{\underline{E}}G

One important contrast between \underline{\underline{E}}G and the more well-known spaces EG and \underline{E}G is the behaviour with respect to direct products. For any groups G and H, one has that E(G×H) = EG×EH and \underline{E}(G×H) = \underline{E}G×\underline{E}H, in the sense that the direct product of any two models for the space on the right is a model for the space on the left. For \underline{E}(G × H), this relies on the observation that any finite subgroup of G × H is contained in a unique minimal product G′ × H′, where G′ ≤ G and H′ ≤ H are finite subgroups. The following example shows that \underline{\underline{E}}(−) does not preserve direct products.

Example 3. (A model for \underline{\underline{E}}G, when G = ℤ × ℤ, the direct product of two infinite cyclic groups.) Let T be a countably infinite tree, with vertex set V, and choose any bijection ϕ between V and the set of infinite cyclic quotients of G. Let G act trivially on T, and make a G-CW-complex X together with a G-equivariant projection onto T by the following recipe. Above the vertex v, take a 1-dimensional model for Eϕ(v). Above the edge (v, w) take the join of the chosen models for Eϕ(v) and Eϕ(w), with the map to the interval induced by the projection Eϕ(v)×Eϕ(w)×I → I. Now G acts on X with all stabilizers cyclic, and it may be checked easily that X is a model for \underline{\underline{E}}G. Moreover, the projection from X to T is G-equivariant for the trivial action of G on T. The quotient X/G may also be described via its projection onto T. Above the vertex v is a copy of S^{1}, which is a model for Bϕ(v), and above the edge (v, w) is a copy of S^{3} consisting of the join of the given models for Bϕ(v) and Bϕ(w). In particular, the integral homology groups H_{2}(X/G) and H_{3}(X/G) are both free abelian of infinite rank, and it follows that X is a model for \underline{\underline{E}}G of minimal dimension.

The above example was explained to us by Wolfgang Lück, in the case when T is homeomorphic to the real line. We prefer to give the more general construction to emphasize that the equivariant homeomorphism type of \underline{\underline{E}}G is far from unique, even amongst models that are ‘minimal’ in some sense.

We shall construct a model for \underline{\underline{E}}G, starting from a model for \underline{E}G, for any group G satisfying the conditions listed below.

Proposition 5. The following conditions on a discrete group G are equivalent:

(1) There is a G-set X in which all stabilizers are virtually cyclic and such that each infinite virtually cyclic subgroup H ≤ G fixes exactly one point of X;

(2) there is a G-set X in which all stabilizers are infinite virtually cyclic and such that each infinite virtually cyclic subgroup H ≤ G fixes exactly one point of X;

(3) every infinite virtually cyclic subgroup H ≤ G is contained in a unique maximal virtually cyclic subgroup, which is equal to its own normalizer; (4) every infinite virtually cyclic subgroup H ≤ G is contained in

a unique maximal virtually cyclic subgroup, and the normalizer of each infinite virtually cyclic subgroup is virtually cyclic.

Proof. For any G-set Y, and y ∈ Y, denote by G_{y} the stabilizer of y, and for H ≤ G let Y^{H} denote the set of H-fixed points in Y. Clearly (2) implies (1). Conversely, if X is a G-set satisfying (1), then X′ = {x ∈ X : |G_{x}| = ∞} is a G-set satisfying (2). For (2) ⟹ (3): if X is a G-set satisfying (2), then each G_{x} must be maximal virtually cyclic, since if H ≥ G_{x}, then X^{H} ⊆ X^{G_{x}} = {x}, and so either H = G_{x} or H is not virtually cyclic. Now if C is infinite virtually cyclic, then X^{C} = {x}, and hence C ≤ G_{y} if and only if x = y. Thus every such C is contained in a unique G_{x}, and so these are all of the maximal infinite virtually cyclic subgroups. It remains to show that each maximal infinite virtually cyclic group is equal to its own normalizer. For any H ≤ G, we have that

(G/H)^{H} = {gH : g^{−1}Hg ⊆ H} ⊇ N_{G}(H)/H,

where N_{G}(H) denotes the normalizer of H. But if H is a maximal infinite virtually cyclic subgroup of G, we know that X contains an orbit isomorphic to G/H. Hence we see that

1 = |X^{H}| ≥ |(G/H)^{H}| ≥ |N_{G}(H) : H| ≥ 1,

and so H = N_{G}(H).

For (3) ⟹ (2): let X be the set of maximal infinite virtually cyclic subgroups of G, with G-action given by conjugation. For H ∈ X, we have that G_{H} = N_{G}(H) = H, and so all point stabilizers in X are infinite virtually cyclic. If C is any infinite virtually cyclic group, then there exists a unique H ∈ X with C ≤ H. But

X^{C} = {H ∈ X : C ≤ N_{G}(H)} = {H ∈ X : C ≤ H},

and so |X^{C}| = 1. It is easy to see that (4) ⟹ (3). For (3) ⟹ (4), if C is infinite virtually cyclic, and H is the unique maximal virtually cyclic containing C, then for any g ∈ G one has that g^{−1}Cg ≤ g^{−1}Hg. Hence N_{G}(C) ≤ N_{G}(H), and so N_{G}(C) is virtually cyclic. □

Proposition 6. Suppose that G has the property that every infinite subgroup H that is not virtually cyclic contains a non-abelian free subgroup. Then G satisfies the conditions of Proposition 5.

Remark 7. Gromov hyperbolic groups have this property ([15], theorem 8.37).

Proof. We shall show that condition (4) is satisfied. Suppose that H_{1} ≤ H_{2} ≤ ⋯ is an ascending chain of virtually cyclic subgroups of G. The union H = ⋃_{n} H_{n} does not contain any non-abelian free groups, and so must be virtually cyclic. It follows that every virtually cyclic subgroup of G is contained in a maximal virtually cyclic subgroup.

Now suppose that H is an infinite cyclic subgroup of G, and that H is contained in two maximal virtually cyclic subgroups, K_{1} and K_{2}. For i = 1, 2, let H_{i} be the intersection of the conjugates in K_{i} of H. Then H_{i} is a finite-index normal subgroup of K_{i}, and H′ = H_{1} ∩ H_{2} is an infinite cyclic group which is normal in the subgroup L generated by K_{1} and K_{2}. It remains to show that L must be virtually cyclic, in which case L = K_{1} = K_{2} by maximality of K_{i}. If not, then L must contain a non-abelian free group. In this case L/H′ contains an infinite cyclic subgroup, and hence L contains a subgroup isomorphic to ℤ^{2}, contradiction.

Similarly, if N = N_{G}(K) is the normalizer of an infinite virtually cyclic subgroup, K, let H_{1} be an infinite cyclic subgroup of K, and let H be the intersection of the conjugates in N of H_{1}. Now H is normal in N, and if N is not virtually cyclic, then N contains a non-abelian free group, so N/H contains an infinite cyclic group, and hence N contains ℤ^{2}, giving a contradiction. □

Proposition 8. Suppose that G satisfies condition (1) of Proposition 5, and that E is any model for \underline{E}G. Then the join Y = E ∗ X is a model for \underline{\underline{E}}G.

Proof. Let H be a subgroup of G. If H is finite, then E^{H} is contractible, and so Y^{H} is also contractible. If H is infinite and virtually cyclic, then E^{H} is empty, but X^{H} is a single point and so Y^{H} is a single point. If H is not virtually cyclic, then H fixes no point of E or of X, and hence Y^{H} is empty. □

4. Finiteness conditions

A G-CW-complex is said to be finite if it has finitely many orbits of cells, and of finite type if it has finitely many orbits of cells of each dimension. Finiteness conditions for universal spaces have long played a role in group theory: a group G is finitely generated (resp. finitely presented) if and only if there is a model for EG with finite 1-skeleton (resp. finite 2-skeleton).

Proposition 4 shows that there is a finite \underline{E}G when G is virtually cyclic. Moreover, many finiteness conditions for \underline{E}G are equivalent to finiteness conditions on EH for various subgroups H. For example, it is easy to see that G has an \underline{E}G with finite 0-skeleton if and only if G contains finitely many conjugacy classes of finite subgroups. If G satisfies this condition, it may be shown that there is an \underline{E}G with finite n-skeleton if and only if there is an EN with finite n-skeleton, for each N ≤ G that is the normalizer of a finite subgroup of G [10, 20]. (See also [18], where examples are constructed to show that finiteness conditions for \underline{E}G are not preserved by finite-index extensions.)

On the other hand, we know of no group G for which \underline{\underline{E}}G is finite, or even of finite type, except the trivial case when G is virtually cyclic (so that a single point will serve as a model for \underline{\underline{E}}G). The join construction in Proposition 8 will never be finite (except in trivial cases), but it can be used to deduce the following.

Proposition 9. Let G be a group satisfying the conditions of Proposition 5, let C_{o} (resp. C_{n}) be a set of conjugacy class representatives of orientable (resp. non-orientable) maximal infinite virtually cyclic subgroups of G, and let C = C_{o} ∪ C_{n}. For any model E for \underline{E}G, a model for \underline{\underline{E}}G may be made by adding to E:

(1) orbits of 0-cells indexed by C;

(2) orbits of 1-cells indexed by C_{o} ∪ {1, 2} × C_{n};

(3) orbits of 2-cells indexed by C.

Proof. Let E be a model for \underline{E}G, and let X be a G-set satisfying condition (2) of Proposition 5. The join E ∗ X is obtained from the disjoint union of E, X and E × X × I by identifying (e, x, 0) ∈ E × X × I with e ∈ E and (e, x, 1) ∈ E × X × I with x ∈ X, for each e and x. For H ∈ C, let E_{H} denote the model for \underline{E}H described in Proposition 4, and define a G-CW-complex Z by

Z = ∐_{H∈C} G ×_{H} E_{H},

where as usual for an H-space Y, G×_{H} Y denotes the induced G-space. There is an equivariant homotopy equivalence E × X ≃_{G} Z. There is an equivariant projection map π : Z → X, and by the universal property of E = \underline{E}G, there is a unique equivariant homotopy class of map ϕ : Z → E. The required model is made by attaching Z × I to the disjoint union E ∐ X, identifying (z, 0) with ϕ(z) and identifying (z, 1) with π(z), for each z ∈ Z. □

Corollary 10. Let G be as in Proposition 9. A space homotopy equivalent to \underline{\underline{B}}G = \underline{\underline{E}}G/G may be made from \underline{B}G by attaching 2-cells indexed by C_{o}.

Proof. Take all notation as in Proposition 9. Thus there is a model for \underline{\underline{E}}G/G given by attaching Z/G × I to the disjoint union E/G ∐ X/G. Here X/G may be identified with C, and E/G is a model for \underline{B}G. In the case when H is orientable, E_{H}/H is homeomorphic to the circle S^{1}, and in the case when H is non-orientable, E_{H}/H is homeomorphic to the interval. Thus Z/G is homeomorphic to the disjoint union of a collection of intervals indexed by C_{n} and a collection of circles indexed by C_{o}. Attaching a cone on an interval to E/G does not change its homotopy type, and attaching a cone on a circle is equivalent to attaching a 2-cell. □

If G is word hyperbolic, there is a finite model for \underline{E}G, namely the Rips complex R_{d}(G) for large integers d. The fact that R_{d}(G) is a model for \underline{E}G was stated in [5], but the first published proof was given in [22]. We remark in passing that it is possible that there exist values of d for which R_{d}(G) is contractible but is not a model for \underline{E}G.

Theorem 11. Let G be a word hyperbolic group that is not virtually cyclic. Then G contains infinitely many conjugacy classes of maximal infinite virtually cyclic subgroups.

Proof. See Corollary 5.1.B or Corollary 8.2.G of [16]. □

Corollary 12. If G is word hyperbolic and not virtually cyclic, then any model for \underline{\underline{E}} contains infinitely many orbits of cells. There is a model containing only finitely many orbits of cells of dimensions 3 or more.

Theorem 13. Let G be a word hyperbolic group that is not virtually cyclic. Then G contains infinitely many conjugacy classes of orientable maximal infinite virtually cyclic subgroups.

Proof. This is a special case of proposition 2 of [8]. Let S be a finite generating set for G, and let X be the Cayley graph of G with respect to S, a graph with vertex set G and edges the ordered pairs (g, gs) for g ∈ G and s ∈ S. The graph X is δ-hyperbolic, and G acts by isometries on it. The remaining hypotheses follow from Theorem 11. The fact that the conclusions of proposition 2 of [8] imply our conclusion follows from the remarks at the top of page 73 of [8]. □

Remark 14. In the case when G is word hyperbolic and contains no elements of order two, it is easy to see that G cannot contain any nonorientable infinite cyclic subgroups. This gives an easy proof of this case of Theorem 13.

Remark 15. Let G be the group given as an extension with kernel ℤ^{2} and quotient cyclic of order two, where the generator for the quotient acts by sending each element of ℤ to its inverse. In this group G, every infinite cyclic subgroup is non-orientable, although of course this group is not hyperbolic. Models for \underline{\underline{E}}G and \underline{\underline{B}}G may be constructed in the same way as for ℤ^{2}, and it may be shown that \underline{\underline{B}}G is homotopy equivalent to a wedge of countably many 3-spheres.

Corollary 16. Let G be a word hyperbolic group that is not virtually cyclic. Then each of the integral homology groups H_{i}(\underline{\underline{B}}G) is finitely generated, except that H_{2}(\underline{\underline{B}}G) contains a free abelian subgroup of infinite rank.

In view of the above examples we suggest the following.

Conjecture 1. If G is a group for which there is a finite model for \underline{\underline{E}}G, then G is virtually cyclic.

The situtation for \underline{\underline{B}}G is slightly more complicated. If G is a group which is locally virtually cyclic, by which one means that every finitely generated subgroup of G is virtually cyclic, then the techniques used in [13] show that \underline{\underline{B}}G is contractible. As a simple example, consider the case when G is countable and locally virtually cyclic but not itself virtually cyclic. Write G as an ascending union:

G = ⋃_{i≥0} G_{i}, G_{i} ⊆ G_{i+1},

where each G_{i} is virtually cyclic. Let X be the graph whose vertices are the cosets G/G_{i}, with an edge from gG_{i} to hG_{j} if and only if j = i + 1 and gG_{i} ⊆ G_{j}. It is easily shown that X is a model for \underline{\underline{E}}G, and the quotient X/G is homeomorphic to the positive reals.

It may be shown that any torsion-free locally virtually cyclic group embeds in the additive group of the rationals as a subgroup, but examples containing torsion abound. For example, any locally finite group is locally virtually cyclic, as is the direct product of a locally finite group and ℤ.

Discussions with Peter Linnell and Thomas Schick were helpful in the construction of the following:

Example 4. (A group G which is not locally virtually cyclic and such that \underline{\underline{B}}G is contractible.) Let 𝔽 be a countably infinite field of characteristic p > 0, with λ ∈ 𝔽 an element of infinite multiplicative order. Let G be the group of affine transformations whose ‘gradient’ is a power of λ:

G = {x ↦ λ^{i}x + b : i ∈ ℤ, b ∈ 𝔽}.

The subgroup N consisting of the translations x ↦ x + b is normal and is isomorphic to the additive group of 𝔽, with G/N infinite cyclic. Let X = 𝔽, and let G act on X with its natural action. For this X, condition (2) of Proposition 5 may be seen to hold. Since G acts transitively on X, one deduces that G contains a single conjugacy class of maximal infinite cyclic subgroups, and that such subgroups are selfnormalizing.

As N is a countable locally finite group, there is a 1-dimensional model T for \underline{E}N. Let T^{(i)} be a copy of T but with the N-action twisted by conjugation (in G) by λ^{i}. As N-spaces,

G ×_{N} T ≅ ∐_{i∈ℤ}T^{(i)}.

Each T^{(i)} is a model for \underline{E}N, and so there is an N-equivariant map f : T → T^{(1)}, and f : T^{(i)} → T^{(i+1)} is also N-equivariant. Let C(f) be the mapping cylinder for f, with its given N-action, and let i : T → C(f) and j : T^{(1)} → C(f) be the inclusions of the ends of the cylinder. Make a G-CW-complex E by identifying each point of G ×_{N} T with its images under 1_{G} × i and 1_{G} × j. This E is a 2-dimensional model for \underline{E}G, built as the double mapping telescope of the map f : T^{(i)} → T^{(i+1)}. Since T/N is contractible, E/G is homotopy equivalent to the circle S^{1}. By Corollary 10, a model for \underline{\underline{B}}G can be built by attaching a single 2-cell to E/G. Since the fundamental group of \underline{\underline{B}}G is trivial, it follows that \underline{\underline{B}}G must be contractible.

Remark 17. We do not know of any group G for which \underline{\underline{B}}G is homotopy equivalent to a finite complex but is not contractible.

5. Computations

Suppose that H_{∗} is an equivariant generalized homology theory, in the sense of [9]. Suppose also that H_{∗} takes disjoint unions to direct sums. Our model for \underline{\underline{E}}G gives the following result.

Proposition 18. For any G satisfying the conditions of Propostion 5, and for any equivariant homology theory as above, there is a long exact sequence

⋯ → ⨁_{(V)} H_{n}^{V} (\underline{E}V) → H_{n}^{G}(\underline{E}G) ⊕ ⨁_{(V)} H_{n}^{V} (\{pt\}) → H_{n}^{G}(\underline{\underline{E}}G) →

⨁_{(V)} H_{n−1}^{V}(\underline{E}V) → H_{n−1}^{G}(\underline{E}G) ⊕ ⨁_{(V)} H_{n−1}^{V}(\{pt\}) → H_{n−1}^{G}(\underline{\underline{E}}G) →

where the indexing set for the sums is a set of conjugacy class representatives of maximal virtually cyclic subgroups V ≤ G.

Proof. Consider the model for \underline{\underline{E}}G given as the join of \underline{E}G and X, the set of maximal infinite virtually cyclic subgroups of G. If X is empty then \underline{\underline{E}}G = \underline{E}G, and the statement follows easily. If X is nonempty, one may write the join as \underline{E}G × X × I modulo the equivalence relation generated by (e, x, 0) ∼ (e, x′, 0) and (e, x, 1) ∼ (e′, x, 1) for all e, e′ ∈ \underline{E}G and x, x′ ∈ X. Let A be the points of the join expressible as (e, x, t) for some t ≤ 2/3 and let B be the points of the join expressible as (e, x, t) for some t ≥ 1/3. Then A is homotopy equivalent to \underline{E}G, B is homotopy equivalent to X, and A ∩ B is homotopy equivalent to the disjoint union ∐_{V ∈X} \underline{E}V.

For any V -CW-complex Y, there is an induction isomorphism

H_{∗}^{G}(G ×_{V} Y) ≅ H_{∗}^{V} (Y).

This together with the Mayer-Vietoris sequence for \underline{\underline{E}}G = A ∪ B gives the claimed long exact sequence. □

For an associative ring R, let 𝕂_{R} be the equivariant algebraic Ktheory spectrum defined in [12], and for a G-CW-complex X, write H_{∗}^{G}(X; 𝕂_{R}) for the associated equivariant homology theory applied to X. This theory is such that for any H ≤ G, H_{∗}^{G}(G/H; 𝕂_{R}) is naturally isomorphic to the (non-connective) algebraic K-theory of the group ring RH. According to the Farrell-Jones isomorphism conjecture (FJC) for algebraic K-theory, the map \underline{\underline{E}}G → G/G induces an isomorphism

H_{∗}^{G}(\underline{\underline{E}}G; 𝕂_{R}) → H^{∗}(G/G; 𝕂_{R}) = K_{∗}(RG).

This conjecture and its analogue for algebraic L-theory motivate the study of \underline{\underline{E}}G.

Corollary 19. For any G satisfying the conditions of Proposition 5, there is an isomorphism

H_{n}^{G}(\underline{\underline{E}}G; 𝕂_{R}) ≅ H_{n}^{G}(\underline{E}G; 𝕂_{R}) ⊕ ⨁_{(V)} Cok_{n}(V),

where Cok_{n}(V) is defined to be the cokernel of the map H_{n}^{V} (\underline{E}V; 𝕂_{R}) → H_{n}^{V} (\underline{\underline{E}}V; 𝕂_{R}).

Proof. This follows from Proposition 18, together with the fact that whenever H is virtually cyclic, the map H_{n}^{H}(\underline{E}H; 𝕂_{R}) → H_{n}^{H}(\underline{\underline{E}}H; 𝕂_{R}) is injective. In fact, Bartels has shown that for any group H, this map is split injective [1]. □

The groups Cok_{n}(V) occurring in the statement of Corollary 19 can be expressed in terms of Nil groups: possibly twisted versions of Bass’s Nil groups [4] in the case when V is orientable; and Waldhausen’s Nil groups [25] in the case when V is non-orientable. For a description of both sorts of Nil group see [11].

The simplest case is when V is orientable, so that V/F (V) ≅ ℤ, and the conjugation action of V/F (V) on F (V) is trivial. In this case one has an isomorphism

Cok_{n}(V) ≅ NK_{n}(RF (V)) ⊕ NK_{n}(RF (V)),

where NK_{n}(S) is the usual Bass Nil group defined as the cokernel of the split injection K_{n}(S) → K_{n}(S[t]). One situation in which this simplest case happens for all V ≤ G is when G is torsion free.

Corollary 20. Suppose that G is torsion-free and satisfies the conditions of Proposition 5. Then there is an isomorphism

H_{n}^{G}(\underline{\underline{E}}G; 𝕂_{R}) ≅ H_{n}^{G}(\underline{E}G; 𝕂_{R}) ⊕ ⨁_{(V)}(NK_{n}(R) ⊕ NK_{n}(R)).

Proof. The hypotheses imply that each infinite virtually cyclic subgroup V is orientable, and that each F (V) is the trivial group. □

Remark 21. In the case when G is the fundamental group of a closed negatively curved manifold, Corollary 20 is proposition 10.5 of [2]. In this case the Farrell-Jones holds and so H_{n}(\underline{\underline{E}}G; 𝕂_{R}) ≅ K_{n}(RG) (see [2] which covers the case when n ≤ 2 and [3] for general n).

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Instituto de Matemáticas, UNAM Campus Morelia, Apartado Postal 61-3 (Xangari), Morelia, Michoacán, MEXICO 58089

  E-mail address: daniel@matmor.unam.mx

School of Mathematics, University of Southhampton, SO17 1BJ, United Kingdom

  E-mail address: i.j.leary@maths.soton.ac.uk