On the classifying space of the family of virtually cyclic subgroups

Wolfgang Lück and Michael Weiermann^{∗} Fachbereich Mathematik Universität Münster Einsteinstr. 62 48149 Münster Germany

January 7, 2009

Abstract

We study the minimal dimension of the classifying space of the family of virtually cyclic subgroups of a discrete group. We give a complete answer for instance if the group is virtually poly-ℤ, word-hyperbolic or countable locally virtually cyclic. We give examples of groups for which the difference of the minimal dimensions of the classifying spaces of virtually cyclic and of finite subgroups is −1, 0 and 1, and show in many cases that no other values can occur.

Key words: classifying spaces of families, dimensions, virtually cyclic subgroups Mathematics Subject Classification 2000: 55R35, 57S99, 20F65, 18G99.

0 Introduction

In this paper, we study the classifying spaces \underline{\underline{E}}G = E_{VCY}(G) of the family of virtually cyclic subgroups of a group G. We are mainly interested in the minimal dimension hdim^{G}(\underline{\underline{E}}G) that a G-CW -model for \underline{\underline{E}}G can have. The classifying space for proper G-actions \underline{E}G = E_{FIN} (G) has already been studied intensively in the literature. The spaces \underline{E}G and \underline{\underline{E}}G appear in the source of the assembly maps whose bijectivity is predicted by the Baum-Connes conjecture and the Farrell-Jones conjecture, respectively. Hence the analysis of \underline{E}G and \underline{\underline{E}}G is important for computations of K- and L-groups of reduced group C^{∗}-algebras and of group rings (see Section 6). These spaces can also be viewed as invariants of G, and one would like to understand what properties of G are reflected in the geometry and homotopy theoretic properties of \underline{E}G and \underline{\underline{E}}G.

∗email: lueck@math.uni-muenster.de, michi@uni-muenster.de www: http://www.math.uni-muenster.de/u/lueck/ FAX: 49 251 8338370

The main results of this manuscript aim at the computation of the homotopy dimension hdim^{G}(\underline{\underline{E}}G) (see Definition 4.1), which is much more difficult than in the case of hdim^{G}(\underline{E}G). It has lead to some (at least for us) surprising phenomenons, already for some nice groups which can be described explicitly and have interesting geometry.

After briefly recalling the notion of \underline{E}G and \underline{\underline{E}}G in Section 1, we investigate in Section 2 how one can build E_{G}(G) out of E_{F}(G) for families F ⊆ G. We construct a G-pushout

\xymatrix{∐_{V ∈M} G ×_{N_{G}V} \underline{E}N_{G}V \ar[d]^{∐_{V ∈M} id_{G} ×f_{V}} \ar[r]^{i} & \underline{E}G \ar[d] \\ ∐_{V ∈M} G ×_{N_{G}V} EW_{G}V \ar[r] & \underline{\underline{E}}G}

for a certain set M of maximal virtually cyclic subgroups of G, provided that G satisfies the condition (M_{FIN ⊆VCY}) that every infinite virtually cyclic subgroup H is contained in a unique maximal infinite virtually cyclic subgroup H_{max}.

For a subgroup H ⊆ G, we denote by N_{G}H := {g ∈ G | gHg^{−1} = H} its normalizer and put W_{G}H := N_{G}H/H. We will say that G satisfies (NM_{FIN ⊆VCY}) if it satisfies (M_{FIN ⊆VCY}) and N_{G}V = V, i.e., W_{G}V = {1}, holds for all V ∈ M. In Section 3, we give a criterion for G to satisfy (NM_{FIN ⊆VCY}).

Section 4 is devoted to the construction of models for \underline{\underline{E}}G and \underline{E}G from models for \underline{\underline{E}}G_{i} and \underline{E}G_{i}, respectively, if G is the directed union of the directed system {G_{i} | i ∈ I} of subgroups.

In Section 5, we prove various results concerning hdim^{G}(\underline{\underline{E}}G) and give some examples. In Subsection 5.1, we prove

hdim^{G}(\underline{E}G) ≤ 1 + hdim^{G}(\underline{\underline{E}}G).

While one knows that hdim^{G×H}(\underline{E}(G × H)) ≤ hdim^{G}(\underline{E}G) + hdim^{H}(\underline{E}H), we prove that

hdim^{G×H}(\underline{\underline{E}}(G × H)) ≤ hdim^{G}(\underline{\underline{E}}G) + hdim^{H}(\underline{\underline{E}}H) + 3

and give examples showing that this inequality cannot be improved in general. In Subsection 5.2, we show

hdim^{G}(\underline{\underline{E}}G) \begin{cases}= hdim^{G}(\underline{E}G) & if hdim^{G}(\underline{E}G) ≥ 2; \\ ≤ 2 & if hdim^{G}(\underline{E}G) ≤ 1 \end{cases}

provided that G satisfies (NM_{FIN ⊆VCY}), and, more generally,

hdim^{G}(\underline{\underline{E}}G) ≤ hdim^{G}(\underline{E}G) + 1

provided that G satisfies (M_{FIN ⊆VCY}). This implies that the fundamental group π of a closed hyperbolic closed manifold M satisfies

vcd(π) = dim(N) = hdim^{π}(\underline{E}π) = hdim^{π}(\underline{\underline{E}}π),

yielding a counterexample to a conjecture due to Connolly-Fehrmann-Hartglass.

In Subsection 5.3, a complete computation of hdim^{G}(\underline{\underline{E}}G) is presented for virtually poly-ℤ groups. This leads to some interesting examples in Subsection 5.4. For instance, we construct, for k = −1, 0, 1, automorphisms f_{k}: Hei → Hei of the three-dimensional Heisenberg group Hei such that

hdim^{Hei ⋊_{f_{k}} ℤ}(\underline{\underline{E}}(Hei ⋊_{f_{k}}ℤ)) = 4 + k.

Notice that hdim^{Hei ⋊_{f_{k}} ℤ}(\underline{E}(Hei ⋊_{f}ℤ)) = cd(Hei ⋊_{f}ℤ) = 4 holds for any automorphism f: Hei → Hei.

In Subsection 5.5, we briefly investigate hdim^{G}(E_{SFG}(G)), where SFG is the family of subgroups of G which are contained in some finitely generated subgroup. After that, in Subsection 5.6, we deal with groups for which the values of hdim^{G}(\underline{E}G) and hdim^{G}(\underline{\underline{E}}G) are small. We show that a countable infinite group G is locally finite if and only if hdim^{G}(\underline{E}G) = hdim^{G}(\underline{\underline{E}}G) = 1.

Finally, we explain in Section 6 how the models we construct for \underline{\underline{E}}G can be used to obtain information about the relative equivariant homology group that appears in the source of the assembly map in the Farrell-Jones conjecture for algebraic K-theory.

All the results in this paper raise the following question:

For which groups G is it true that

hdim^{G}(\underline{E}G) − 1 ≤ hdim^{G}(\underline{\underline{E}}G) ≤ hdim^{G}(\underline{E}G) + 1 ?

We have no example of a group for which the above inequality is not true.

In this paper, we are not dealing with the question whether there is a finite G-CW -model for \underline{\underline{E}}G. Namely, there is no known counterexample to the conjecture (see [10, Conjecture 1]) that a group G possessing a finite G-CW -model for \underline{\underline{E}}G is virtually cyclic.

The authors wishes to thank the referee for his detailed and valuable report. The paper was supported by the Sonderforschungsbereich 478 – Geometrische

Strukturen in der Mathematik – the Max-Planck-Forschungspreis and the Leibniz-Preis of the first author. Parts of this paper have already appeared in the Ph.D. thesis of the second author.

1 Classifying Spaces for Families

We briefly recall the notions of a family of subgroups and the associated classifying spaces. For more information, we refer for instance to the survey article [16].

A family F of subgroups of G is a set of subgroups of G which is closed under conjugation and taking subgroups. Examples for F are

{1} = {trivial subgroup};
FIN = {finite subgroups};
VCY = {virtually cyclic subgroups};
SFG = {subgroups of finitely generated subgroups};
ALL = {all subgroups}.

Let F be a family of subgroups of G. A model for the classifying space E_{F}(G) of the family F is a G-CW -complex X all of whose isotropy groups belong to F and such that for any G-CW -complex Y whose isotropy groups belong to F there is precisely one G-map Y → X up to G-homotopy. In other words, X is a terminal object in the G-homotopy category of G-CW -complexes whose isotropy groups belong to F. In particular, two models for E_{F}(G) are G-homotopy equivalent, and for two families F_{0} ⊆ F_{1} there is precisely one G-map E_{F_{0}}(G) → E_{F_{1}}(G) up to G-homotopy. There exists a model for E_{F}(G) for any group G and any family F of subgroups.

A G-CW -complex X is a model for E_{F}(G) if and only if the H-fixed point set X^{H} is contractible for H ∈ F and is empty for H ∉ F.

We abbreviate \underline{E}G := E_{FIN} (G) and call it the universal G-CW -complex for proper G-actions. We also abbreviate \underline{\underline{E}}G := E_{VCY}(G).

A model for E_{ALL}(G) is G/G. A model for E_{{1}}(G) is the same as a model for EG, which denotes the total space of the universal G-principal bundle EG → BG. Our interest in these notes concerns the spaces \underline{\underline{E}}G.

2 Passing to larger families

In this section, we explain in general how one can construct a model for E_{G}(G) from E_{F}(G) if F and G are families of subgroups of the group G with F ⊆ G. Let ∼ be an equivalence relation on G \ F with the properties:

• If H, K ∈ G \ F with H ⊆ K, then H ∼ K;

• If H, K ∈ G \ F and g ∈ G, then H ∼ K ⇔ gHg^{−1} ∼ gKg^{−1}. (2.1)

Let [G \ F] be the set of equivalence classes of ∼. Denote by [H] ∈ [G \ F] the equivalence class of H ∈ G \ F, and define the subgroup

N_{G}[H] := {g ∈ G | [g^{−1}Hg] = [H]}

of G. Then N_{G}[H] is the isotropy group of [H] under the G-action on [G \ F] induced by conjugation. Define a family of subgroups of N_{G}[H] by

G[H] := {K ⊆ N_{G}[H] | K ∈ G \ F, [K] = [H]} ∪ (F ∩ N_{G}[H]),

where F ∩ N_{G}[H] consists of all the subgroups of N_{G}[H] belonging to F.

Definition 2.2 (Equivalence relation on VCY \ FIN). If G = VCY and F = FIN, we will take for the equivalence relation ∼

V ∼ W ⇔ |V ∩ W | = ∞.

Theorem 2.3 (Constructing models from given ones). Let F ⊆ G and ∼ be as above such that properties (2.1) hold. Let I be a complete system of representatives [H] of the G-orbits in [G \ F] under the G-action coming from conjugation. Choose arbitrary N_{G}[H]-CW -models for E_{F∩N_{G}[H]}(N_{G}[H]) and E_{G[H]}(N_{G}[H]), and an arbitrary G-CW -model for E_{F}(G). Define a G-CW - complex X by the cellular G-pushout

\xymatrix{∐_{[H]∈I} G ×_{N_{G}[H]} E_{F∩N_{G}[H]}(N_{G}[H]) \ar[d]^{∐_{[H]∈I} id_{G} ×_{N_{G}[H]}f_{[H]}} \ar[r]^{i} & E_{F}(G) \ar[d] \\ ∐_{[H]∈I} G ×_{N_{G}[H]} E_{G[H]}(N_{G}[H]) \ar[r] & X}

such that f_{[H]} is a cellular N_{G}[H]-map for every [H] ∈ I and i is an inclusion of G-CW -complexes, or such that every map f_{[H]} is an inclusion of N_{G}[H]-CW - complexes for every [H] ∈ I and i is a cellular G-map.

Then X is a model for E_{G}(G).

Proof. We have to show that X^{K} is contractible if K belongs to G, and that it is empty, otherwise. For [H] ∈ I, let s_{[H]} : G/N_{G}[H] → G be a set-theoretic section of the projection G → G/N_{G}[H]. Let K ⊆ G be a subgroup. Taking Kfixed points in the above G-pushout yields, up to homeomorphism, the following pushout

\xymatrix{∐_{[H]∈I} ∐_{α} E_{F∩N_{G}[H]}(N_{G}[H])^{s_{[H]}(α)^{−1}Ks_{[H]}(α)} \ar[d]_{∐_{[H]∈I} ∐_{α} f_{[H],α}} \ar[r]^{i} & E_{F}(G)^{K} \ar[d] \\ ∐_{[H]∈I} ∐_{α} E_{G[H]}(N_{G}[H])^{s_{[H]}(α)^{−1}Ks_{[H]}(α)} \ar[r] & X^{K}}  (2.4)

in which one of the maps starting from the left upper corner is a cofibration and α runs through {α ∈ G/N_{G}[H] | s_{[H]}(α)^{−1}Ks_{[H]}(α) ⊆ N_{G}[H]}.

Assume first that K ∉ G. Then the entries in the upper row and the lower left entry of (2.4) are clearly empty. Hence, in this case X^{K} is empty.

If K ∈ G\F, the entries in the upper row of (2.4) are again empty, so in order to show that X^{K} is contractible, we must show that the lower left entry of (2.4) is contractible. Note that the space E_{G[H]}(N_{G}[H])^{s_{[H]}(α)^{−1}Ks_{[H]}(α)} is non-empty if and only if s_{[H]}(α)^{−1}Ks_{[H]}(α) belongs to the family G[H] of subgroups of N_{G}[H], which is equivalent to the condition that [s_{[H]}(α)^{−1}Ks_{[H]}(α)] = [H] and s_{[H]}(α)^{−1}Ks_{[H]}(α) ⊆ N_{G}[H]. Now, there is precisely one [H_{K}] ∈ I for which there exists g_{K} ∈ G with the property that [g_{K}^{−1}Kg_{K}] = [H_{K}]. An element g ∈ G satisfies [g^{−1}Kg] = [H_{K}] if and only if g_{K}^{−1}g ∈ N_{G}[H_{K}]. Moreover, L ⊆ N_{G}[H] always holds if L ∈ G \ F is such that [L] = [H]. We conclude from these observations that E_{G[H]}(N_{G}[H])^{s_{[H]}(α)^{−1}Ks_{[H]}(α)} is non-empty if and only if [H] = [H_{K}] and α = g_{K}N_{G}[H_{K}]. Furthermore, it is contractible in this case. This implies that the lower left entry of (2.4) is contractible.

Finally, if K ∈ F, then the upper right entry of (2.4) is contractible. This implies that the same will hold for X^{K} if we can show that all the maps f_{[H],α} are homotopy equivalences. But this is clear since the source and target spaces of the f_{[H],α} are contractible.

Remark 2.5 (Dimension of the model constructed for E_{G}(G)). A Gpushout as described in Theorem 2.3 always exists. To be more precise, the maps i and f_{[H]} exist and are unique up to equivariant homotopy equivalence due to the universal property of the classifying spaces E_{F}(G) and E_{G[H]}(N_{G}[H]). Moreover, because of the equivariant cellular approximation theorem (see for instance [13, Theorem 2.1 on page 32]), these maps can be assumed to be cellular. Finally, one can replace a cellular G-map f: X → Y by the canonical inclusion j: X → cyl(f) into its mapping cylinder. Then the canonical projection p: cyl(f) → Y is a G-homotopy equivalence such that pr ∘i = f.

Note that the dimension of cyl(f) is max{1 + dim(X), dim(Y)}. This means that we we get the following conclusion from Theorem 2.3 which we will often use: There exists an n-dimensional G-CW -model for E_{G}(G) if there exists an n-dimensional G-CW -model for E_{F}(G) and, for every H ∈ I, an (n − 1)- dimensional N_{G}[H]-CW -model for E_{F∩N_{G}[H]}(N_{G}[H]) and an n-dimensional N_{G}[H]-CW -model for E_{G[H]}(N_{G}[H]).

Example 2.6. In general N_{G}[V], cannot be written as a normalizer N_{G}W for some W ∈ [V] in the case G = VCY, F = FIN and ∼ as defined in Definition 2.2.

For instance, let p be a prime number. Let ℤ[1/p] be the subgroup of ℚ consisting of rational numbers x ∈ ℚ for which p^{n} · x ∈ ℤ for some positive integer n. This is the directed union ⋃_{n∈ℕ} p^{−n} · ℤ. Let p · id: ℤ[1/p] → ℤ[1/p] be the automorphism given by multiplication with p, and define G to be the semi-direct product ℤ[1/p] ⋊_{p·id} ℤ. Let C be the cyclic subgroup of ℤ[1/p] generated by 1. Then N_{G}[C] = G but N_{G}V = ℤ[1/p] for every V ∈ [C].

Notation 2.7. Let F ⊆ G be families of subgroups of a group G.

We shall say that G satisfies (M_{F⊆G}) if every subgroup H ∈ G \ F is contained in a unique H_{max} ∈ G \ F which is maximal in G \ F, i.e., H_{max} ⊆ K for K ∈ G \ F implies H_{max} = K.

We shall say that G satisfies (NM_{F⊆G}) if M satisfies (M_{F⊆G}) and every maximal element M ∈ G \ F equals its normalizer, i.e., N_{G}M = M or, equivalently, W_{G}M = {1}.

Examples of groups satisfying (NM_{FIN ⊆VCY}) will be given in Section 3. In [7, page 101], Davis-Lück discuss examples of groups satisfying (M_{{1}⊆FIN}) are discussed. The following will be a consequence of Theorem 2.3:

Corollary 2.8. Let F ⊆ G be families of subgroups of a group G which satisfies (M_{F⊆G}). Let M be a complete system of representatives of the conjugacy classes of subgroups in G \ F which are maximal in G \ F. Let SUB(M) be the family of subgroups of M. Consider a cellular G-pushout

\xymatrix{∐_{M∈M} G ×_{N_{G}M} E_{F∩N_{G}M}(N_{G}M) \ar[d]^{∐_{M∈M} id_{G} ×_{N_{G}M} f_{[H]}} \ar[r]^{i} & E_{F}(G) \ar[d] \\ ∐_{M∈M} G ×_{N_{G}M} E_{SUB(M)∪(F∩N_{G}M)}(N_{G}M) \ar[r] & X}

such that f_{[H]} is a cellular N_{G}[H]-map for every [H] ∈ I and i is an inclusion of G-CW -complexes, or such that f_{[H]} is an inclusion of N_{G}[H]-CW -complexes for every [H] ∈ I and i is a cellular G-map.

Then X is a model for E_{G}(G).

Proof. Let H_{max} denote the unique maximal element in G \ F which contains H ∈ G \F. We use the equivalence relation on G \F given by H ∼ K ⇔ H_{max} = K_{max}. Then, for M ∈ M, the family G[M] is just {K ∈ G | K ⊆ M or K ∈ F}. Now apply Theorem 2.3.

Remark 2.9. If G satisfies (M_{FIN ⊆VCY}), then the equivalence relation ∼ defined in Definition 2.2 agrees with the equivalence relation H ∼ K ⇔ H_{max} = K_{max} appearing in the proof of Corollary 2.8 above.

Corollary 2.10. Let G be a group satisfying (M_{{1}⊆FIN}) or (M_{FIN ⊆VCY}) respectively. We denote by M a complete system of representatives of the conjugacy classes of maximal finite subgroups F ⊆ G or of maximal infinite virtually cyclic subgroups V ⊆ G respectively. Consider the cellular G-pushouts

\xymatrix{∐_{F ∈M} G ×_{N_{G}F} EN_{G}F \ar[dd]^{∐_{F ∈M} id_{G} ×f_{F}} \ar[r]^{i} & EG \ar[dd] &  & ∐_{V ∈M} G ×_{N_{G}V} \underline{E}N_{G}V \ar[dd]^{∐_{V ∈M} id_{G} ×f_{V}} \ar[r]^{i} & \underline{E}G \ar[dd] \\  &  & or \\ ∐_{F ∈M} G ×_{N_{G}F} EW_{G}F \ar[r] & X &  & ∐_{V ∈M} G ×_{N_{G}V} EW_{G}V \ar[r] & Y}

where EW_{G}H is viewed as an N_{G}H-CW -complex by restricting with the projection N_{G}H → W_{G}H for H ⊆ G, the maps starting from the left upper corner are cellular and one of them is an inclusion of G-CW -complexes.

Then X is a model for \underline{E}G or Y is a model for \underline{\underline{E}}G respectively.

Proof. This follows from Corollary 2.8 and the following facts. A model for E_{SUB(H)}(N_{G}H) is given by EW_{G}H considered as an N_{G}H-CW -complex by restricting with the projection N_{G}H → W_{G}H for every H ⊆ G. Furthermore, we obviously have {1} ∩ N_{G}F ⊆ SUB(F) for every maximal finite subgroup F, whereas FIN ∩ N_{G}V ⊆ SUB(V) for every maximal infinite virtually cyclic subgroup V ⊆ G because W_{G}V is torsionfree (if it were not, then the preimage of a non-trivial finite subgroup of W_{G}V under the projection N_{G}V → W_{G}V would be virtually cyclic and strictly containing V, contradicting its maximality).

As a special case of Corollary 2.10, we get

Corollary 2.11. Let G be a group satisfying (NM_{{1}⊆FIN}) or (NM_{FIN ⊆VCY}) respectively. Let M be a complete system of representatives of the conjugacy classes of maximal finite subgroups F ⊆ G or of maximal infinite virtually cyclic subgroups V ⊆ G respectively. Consider the cellular G-pushouts

\xymatrix{∐_{F ∈M} G ×_{F} EF \ar[dd]^{∐_{F ∈M} p} \ar[r]^{i} & EG \ar[dd] &  & ∐_{V ∈M} G ×_{V} \underline{E}V \ar[dd]^{∐_{V ∈M} p} \ar[r]^{i} & \underline{E}G \ar[dd] \\  &  & or \\ ∐_{F ∈M} G/F \ar[r] & X &  & ∐_{V ∈M} G/V \ar[r] & Y}

where i is an inclusion of G-CW -complexes and p is the obvious projection. Then X is a model for \underline{E}G or Y is a model for \underline{\underline{E}}G respectively.

3 A Class of Groups satisfying (NM_{FIN ⊆VCY})

The following provides a criterion for a group to satisfy (NM_{F⊆VCY}):

Theorem 3.1. Suppose that the group G satisfies the following two conditions:

• Every infinite cyclic subgroup C ⊆ G has finite index [C_{G}C : C] in its centralizer;

• Every ascending chain H_{1} ⊆ H_{2} ⊆ … of finite subgroups of G becomes stationary, i.e., there is an n_{0} ∈ ℕ such that H_{n} = H_{n_{0}} for all n ≥ n_{0}.

Then every infinite virtually cyclic subgroup V ⊆ G is contained in a unique maximal virtually cyclic subgroup V_{max} ⊆ G. Moreover, V_{max} is equal to its normalizer N_{G}(V_{max}), and

V_{max} = ⋃_{C⊆V} N_{G}C,

where the union is over all infinite cyclic subgroups C ⊆ V that are normal in V.

In particular, G satisfies (NM_{FIN ⊆VCY}).

Proof. We fix an infinite virtually cyclic subgroup V ⊆ G and define V_{max} := ⋃_{C⊆V} N_{G}C, where the union is over all infinite cyclic subgroups C ⊆ V that are normal in V. The collection of all such subgroups of V is countable, and we denote it by {C_{n}}_{n∈ℕ}. Since every index [V : C_{n}] is finite, [V : C_{0} ∩ … ∩ C_{n}] must also be finite for n ∈ ℕ. Thus, if we set Z_{n} := C_{0} ∩ … ∩ C_{n}, then Z_{n} ⊆ C_{n}, and Z_{0} ⊇ Z_{1} ⊇ … is a descending chain of infinite cyclic subgroups of V that are normal in V. If C′ ⊆ C are two infinite cyclic subgroups of G, then N_{G}C ⊆ N_{G}C′ because C′ is a characteristic subgroup of C. It follows in our situation that N_{G}C_{n} ⊆ N_{G}Z_{n}, which implies

⋃_{n=0}^{∞} N_{G}C_{n} = ⋃_{n=0}^{∞} N_{G}Z_{n}.  (3.2)

Furthermore, N_{G}Z_{0} ⊆ N_{G}Z_{1} ⊆ … is an ascending chain, which becomes stationary by the following argument. Namely, we can estimate

[N_{G}Z_{n} : N_{G}Z_{0}] ≤ [N_{G}Z_{n} : C_{G}Z_{0}] = [N_{G}Z_{n} : C_{G}Z_{n}] · [C_{G}Z_{n} : C_{G}Z_{0}],

and the first factor on the right is not greater than 2 since there is an injection N_{G}Z_{n}/C_{G}Z_{n} → aut(Z_{n}), whereas the second does not exceed an appropriate constant as we will show in Lemma 3.3 below. Hence, it follows from (3.2) that V_{max} = N_{G}Z_{n} for all n ≥ n_{0} if n_{0} ∈ ℕ is sufficiently large. In other words, we can record that this construction yields, for any infinite cyclic subgroup C ⊆ V that is normal in V, an infinite cyclic subgroup Z ⊆ C that is normal in V such that for all infinite cyclic subgroups Z′ ⊆ Z we have V_{max} = N_{G}Z′.

Using the equality V_{max} = N_{G}Z_{n_{0}}, it is now obvious that V_{max} is virtually cyclic since the finite index subgroup C_{G}Z_{n_{0}} is so due to the assumption imposed on G. In addition, V ⊆ V_{max} since Z_{n_{0}} is normal in V. Now suppose W ⊆ G is an infinite virtually cyclic subgroup such that V ⊆ W. We claim that V_{max} = W_{max}. In order to prove this, let Z_{W} ⊆ W be an infinite cyclic subgroup that is normal in W such that W_{max} = N_{G}Z′_{W} holds for all infinite cyclic subgroups Z′_{W} ⊆ Z_{W}. Then Z_{W} ∩ V is a finite index subgroup of W and so is an infinite cyclic subgroup of V that is normal in V. As we have already seen, there is an infinite cyclic subgroup Z_{V} ⊆ Z_{W} ∩ V that is normal in V such that V_{max} = N_{G}Z′_{V} holds for all infinite cyclic subgroups Z′_{V} ⊆ Z_{V}. Since V has finite index in W and Z_{V} is normal in V, the intersection Z′′_{V} of all the conjugates of Z_{V} by elements in W has finite index in Z_{V} and therefore is an infinite cyclic subgroup of Z_{V}. This implies V_{max} = N_{G}Z′′_{V} = W_{max}. From this we can deduce immediately that V_{max} is indeed maximal among virtually cyclic subgroups of G containing V and that it is uniquely determined by this property.

Finally, we will show that N_{G}V_{max} is virtually cyclic, so that it is equal to V_{max}. Let C ⊆ V_{max} be infinite cyclic. Since C has finite index in V_{max} and V_{max} contains only finitely many subgroups of index [V_{max} : C], the group D which we define as the intersection of all conjugates of C in N_{G}V_{max} has finite index in V_{max} and is therefore infinite cyclic as well. Obviously, D is normal in N_{G}V_{max}, so N_{G}V_{max} ⊆ N_{G}D holds, the latter being virtually cyclic because the finite index subgroup C_{G}D is so by assumption. Hence Theorem 3.1 is proven as soon as we have finished the proof of the next lemma.

Lemma 3.3. There exists a natural number N such that for all n ∈ ℕ we have

[C_{G}Z_{n} : C_{G}Z_{0}] ≤ N.

Proof. Since the center of the virtually cyclic group C_{G}Z_{n} contains Z_{n} and hence is infinite, C_{G}Z_{n} is of type I, i.e., possesses an epimorphism onto ℤ. Denoting by T_{n} the torsion subgroup of H_{1}(C_{G}Z_{n}), it then follows that H_{1}(C_{G}Z_{n})/T_{n} is infinite cyclic. Let p_{n}: C_{G}Z_{n} → H_{1}(C_{G}Z_{n}) → H_{1}(C_{G}Z_{n})/T_{n} be the canonical projection, and let c_{n} ∈ C_{G}Z_{n} be such that p_{n}(c_{n}) is a generator.

Consider the commutative diagram

\xymatrix{1 \ar[r] & ker(p_{0}) \ar[d] \ar[r] & C_{G}Z_{0} \ar[d] \ar[r]^{p_{0}} & H_{1}(C_{G}Z_{0})/T_{0} \ar[d] \ar[r] & 1 \\ 1 \ar[r] & ker(p_{1}) \ar[d] \ar[r] & C_{G}Z_{1} \ar[d] \ar[r]^{p_{1}} & H_{1}(C_{G}Z_{1})/T_{1} \ar[d] \ar[r] & 1 \\  & ⋮ & ⋮ & ⋮}

which has exact rows and in which all the vertical arrows in the first and second column are inclusions and all vertical arrows in the third column are induced by the obvious inclusions. They, too, are injective because ker(p_{n}) is finite and H_{1}(C_{G}Z_{n})/T_{n} is infinite cyclic for n ∈ ℕ. Since all the ker(p_{n}) are finite, ⋃_{n=0}^{∞} ker(p_{n}) is again finite, say of order a ∈ ℕ, by the assumption on G. Then in particular [ker(p_{n}) : ker(p_{0})] ≤ a for n ∈ ℕ, so {[C_{G}Z_{n} : C_{G}Z_{0}] | n ∈ ℕ} will be bounded if we can show that the index of H_{1}(C_{G}Z_{0})/T_{0} in H_{1}(C_{G}Z_{n})/T_{n} is less than a constant which does not depend on n ∈ ℕ.

In order to construct such a constant, let r_{n} ∈ ℤ be such that p_{0}(c_{0}) is mapped to p_{n}(c_{n})^{r_{n}} under the inclusions in the above diagram. Then, by exactness, there is a k_{n} ∈ ker(p_{n}) such that c_{0} = k_{n}c_{n}^{r_{n}}. Since the order of ker(p_{n}) divides a, the group aut(ker(p_{n})) contains at most a! elements, so that any ϕ_{n} ∈ aut(ker(p_{n})) satisfies ϕ_{n}^{a!} = id. Put d := a · a!, then we get for every ϕ_{n} ∈ aut(ker(p_{n})) and k ∈ ker(p_{n}) the equality

∏_{i=0}^{d−1} ϕ_{n}^{i}(k) = ∏_{j=1}^{a} ∏_{i=(j−1)·a!}^{j·a!−1} ϕ_{n}^{i}(k) = ( ∏_{i=0}^{a!−1} ϕ_{n}^{i}(k))^{a} = 1.

Specializing to ϕ_{n}(k) := c_{n}^{r_{n}}kc_{n}^{−r_{n}} yields

c_{0}^{d} = (k_{n}c_{n}^{r_{n}})^{d} = (∏_{i=0}^{d−1} ϕ_{n}^{i}(k_{n})) · c_{n}^{r_{n}d} = c_{n}^{r_{n}d}.

If z_{0} is a generator of Z_{0}, then there exists an s ∈ ℤ such that p_{0}(z_{0}) = p_{0}(c_{0})^{s}. This means z_{0}^{−1}c_{0}^{s} ∈ ker(p_{0}), hence z_{0}^{d} = c_{0}^{sd}. Altogether this implies that if dZ_{0} denotes the cyclic group generated by z_{0}^{d}, we have dZ_{0} = ⟨c_{n}^{r_{n}ds}⟩ and thus c_{n} ∈ C_{G}dZ_{0}.

We can finally define b := [C_{G}dZ_{0} : dZ_{0}], which is finite by assumption and constitutes the required constant. This is due to the fact that c_{n}^{b} ∈ dZ_{0}, so there is a t_{n} ∈ ℤ such that c_{n}^{b} = c_{n}^{r_{n}dst_{n}}. Hence r_{n} divides b, and the index of H_{1}(C_{G}Z_{0})/T_{0} in H_{1}(C_{G}Z_{n})/T_{n} equals |r_{n}| by construction.

Lemma 3.4. Let G be a group with the property that every non-virtually cyclic subgroup of G contains a copy of ℤ∗ℤ. Then G satisfies the conditions appearing in Theorem 3.1.

Proof. Let C ⊆ G be infinite cyclic. By assumption, its centralizer C_{G}C is either virtually cyclic or contains ℤ ∗ ℤ. To prove the first condition appearing in Theorem 3.1, we have to show that C_{G}C does not contain ℤ ∗ ℤ. Suppose it does. Then ℤ ∗ ℤ ∩ C = {1} as ℤ ∗ ℤ has a trivial center. Hence one of the generators of ℤ ∗ ℤ together with a generator of C generate a copy of ℤ ⊕ ℤ inside G, which contradicts the assumption imposed on G.

It is obvious that any ascending chain H_{1} ⊆ H_{2} ⊆ … of finite subgroups of G must become stationary since, otherwise, ⋃_{n} H_{n} would be an infinite torsion subgroup of G, contradicting the assumptions on G. This proves the second condition appearing in Theorem 3.1.

Remark 3.5. Lemma 3.4 implies that the main result of Juan-Pineda and Leary in [10] is covered by Theorem 3.1. Obviously this is also true for [16, Theorem 8.11] which motivated Theorem 3.1.

It follows from the Kurosh subgroup theorem (see for example [17, Theorem I.5.14 on page 56]) that the class of groups satisfying the conditions of Theorem 3.1 is closed under arbitrary free products, whereas this is not the case for the class of groups considered in [16, Theorem 8.11].

Example 3.6 (Word-hyperbolic groups). Each word-hyperbolic group G satisfies the two conditions appearing in Theorem 3.1. A proof for this can be found in [3, Theorem 3.2 in III.Γ.3 on page 459 and Corollary 3.10 in III.Γ.3 on page 462].

Example 3.7 (A group not satisfying (M_{FIN ⊆VCY})). Let G be the semidirect product ℤ ⋊ ℤ with respect to the group homomorphism ℤ → aut(ℤ) sending the generator to − id: ℤ → ℤ. A presentation of G is given by ⟨s, t | sts^{−1} = t^{−1}⟩. Consider the infinite cyclic subgroups ⟨s⟩ and ⟨st⟩. They are maximal among infinite virtually cyclic subgroups. These two groups are obviously different. Their intersection is the proper subgroup ⟨s^{2}⟩. Hence G does not satisfy (M_{FIN ⊆VCY}).

4 Models for E_{F}(G) for colimits of groups

In this section, we want to construct models for E_{F}(G) if G is a colimit of a directed system of groups {G_{i} | i ∈ I}.

Given a directed set I = (I, ≤), we consider I as a category. Namely, the set of objects is the set I itself and mor_{I}(i, j) consists of precisely one element if i ≤ j and is empty otherwise. A directed system of groups is a covariant functor from the category I into the category of groups, i.e., it consists of a collection {G_{i} | i ∈ I} of groups together with group homomorphisms ϕ_{i,j}: G_{i} → G_{j} for i ≤ j. Its colimit G := colim_{i∈I} G_{i} is a group G together with group homomorphisms ψ_{i}: G_{i} → G for every i ∈ I which satisfy ψ_{j} = ψ_{i} ∘ ϕ_{i,j} for i, j ∈ I and i ≤ j such that the following universal property holds: Given a group K together with group homomorphisms μ_{i}: G_{i} → K satisfying μ_{j} = μ_{i} ∘ ϕ_{i,j} for i, j ∈ I and i ≤ j, there is precisely one group homomorphism ν: G → K satisfying ν ∘ ψ_{i} = μ_{i} for every i ∈ I. If {G_{i} | i ∈ I} is a directed system of subgroups of G directed by inclusion, then G = colim_{i∈I} G_{i} if and only if G = ⋃_{i∈I} G_{i}.

Given a homomorphism ψ: G′ → G of groups and a family of subgroups F of G, define ψ^{∗}F to be the family of subgroups of G′ given by {H ⊆ G′ | ψ(H) ∈ F}. If ψ is an inclusion of groups, then ψ^{∗}F agrees with the family F ∩ G′ consisting of all the subgroups of G′ which belong to F.

Definition 4.1 (Homotopy dimension). Given a G-space X, the homotopy dimension hdim^{G}(X) ∈ {0, 1, …} ⨿ {∞} of X is defined to be the infimum over the dimensions of all G-CW -complexes Y which are G-homotopy equivalent to X.

Obviously, hdim^{G}(X) depends only on the G-homotopy type of X. In particular, hdim^{G}(E_{F}(G)) is well-defined for any group G and any family F. For example, hdim^{G}(E_{F}(G)) = 0 is equivalent to the condition that F = ALL.

Let C be a small category. A contravariant C-CW -complex is a contravariant functor from C to the category of CW -complexes which is built of cells of the shape mor_{C}(?, c) × D^{n}. For a rigorous definition of this object, we refer for instance to [6, Definition 3.2] (where it is called a contravariant free C-CW - complex). A model for the contravariant classifying space EC of the category C is, by definition, a contravariant C-CW -complex whose evaluation at any object in C is contractible. Let hdim(C) be the infimum of the dimensions of all models for EC. The elementary proof of the next result is left to the reader (see [6, Example 3.9])).

Lemma 4.2. Let I be a directed set. Then I contains a maximal element if and only if hdim(I) = 0. If I contains a countable cofinal subset I′, then hdim(I) ≤ 1. (Cofinal means that for every i ∈ I there exists i′ ∈ I with i ≤ i′.)

Now we can formulate the main result of this section, whose proof is carried out in the remainder of this section.

Theorem 4.3. Let {G_{i} | i ∈ I} be a directed system of groups (with not necessarily injective structure maps ϕ_{i,j}: G_{i} → G_{j} for i ≤ j). Let G = colim_{i∈I} G_{i} be the colimit of this directed system. Let F be a family of subgroups of G such that every H ∈ F is contained in im(ψ_{i}) for some i ∈ I, where ψ_{i}: G_{i} → G is the structure map of the colimit for i ∈ I. Then

hdim^{G}(E_{F}(G)) ≤ hdim(I) + sup { hdim^{G_{i}}(E_{ψ_{i}^{∗}F}(G_{i}))|i ∈ I}.

We need some preparations for the proof of Theorem 4.3. In the following, we will use the notation and results of [6, Section 1 and 3], except that we replace the notation ⊗_{C} in [6, Definition 1.4] by ×_{C}. We also mention that, for a covariant functor X: C → SPACES, the space EC ×_{C} X is a model for the homotopy colimit of X (see [6, page 225]).

Lemma 4.4. Let C be a small category. Let Z be a contravariant C-CW - complex and let X: C → G-SPACESbe a covariant functor from C to the category of G-spaces.

Then Z ×_{C} X is a G-space and we have:

(i) If X(c) is a G-CW -complex for every c ∈ C, then Z×_{C}X has the homotopy type of a G-CW -complex. We have

hdim^{G}(Z ×_{C} X) ≤ dim(Z) + sup { hdim^{G}(X(c)) | c ∈ ob(C)};

(ii) For every H ⊆ G there is a natural homeomorphism of spaces

Z ×_{C} X^{H} ⟶^{≅} (Z ×_{C} X)^{H} .

Proof. (i) Let Z_{n} be the n-skeleton of Z_{n}. We construct, by induction over n = −1, 0, 1, …, a sequence of inclusions of G-CW -complexes Y_{−1} ⊆ Y_{0} ⊆ Y_{1} ⊆ … together with G-homotopy equivalences f_{n}: Z_{n} ×_{C} X → Y_{n} such that f_{n}|_{Z_{n−1}×_{C}X} = f_{n−1} and

dim(Y_{n}) ≤ n + sup { hdim^{G}(X(c)) | c ∈ ob(C)}

holds for n = 0, 1, 2, … Then the claim is obviously true if Z is finite-dimen-sional. For infinite-dimensional Z, a standard colimit argument implies that the system (f_{n})_{n∈ℕ} of G-homotopy equivalences yields a G-homotopy equivalence f: Z ×_{C} X → Y := ⋃_{n∈ℕ} Y_{n} since every map Z_{n−1} ×_{C} X → Z_{n} ×_{C} X is a G-cofibration and Z ×_{C} X is ⋃_{n∈ℕ} Z_{n} ×_{C} X_{n} with respect to the weak topology. As Y is a G-CW -complex, the claim follows in this case as well.

The induction begins by taking Y_{−1} = ∅. The induction step from n − 1 to n ≥ 0 is done as follows. We can write Z_{n} as a pushout of contravariant C-spaces

\xymatrix{∐_{j∈J} mor_{C}(?, c_{j}) × S^{n−1} \ar[d]_{∐_{j∈J} id_{mor_{C} (?,c_{j})} ×i} \ar[r] & Z_{n−1} \ar[d] \\ ∐_{j∈J} mor_{C}(?, c_{j}) × D^{n} \ar[r] & Z_{n}}

where i: S^{n−1} → D^{n} is the inclusion. Applying − ×_{C} X yields a G-pushout

\xymatrix{∐_{j∈J} X(c_{j}) × S^{n−1} \ar[d]_{∐_{j∈J} id_{X(c_{j})} ×i} \ar[r]^{u} & Z_{n−1} ×_{C} X \ar[d] \\ ∐_{j∈J} X(c_{j}) × D^{n} \ar[r] & Z_{n} ×_{C} X}

Obviously, the left vertical arrow is a G-cofibration and hence the same is true for the right vertical arrow. Choose, for every c ∈ C, a G-CW -complex X(c)′ of dimension hdim^{G}(X(c)) together with a G-homotopy equivalence u_{c}: X(c)′ → X(c). By the equivariant cellular approximation theorem (see for instance [13, Theorem 2.1 on page 32]), the composition

∐_{j∈J} X(c_{j})′ × S^{n−1} \underrightarrow{∐_{j∈J} u_{c_{j}} ×id_{S^{n−1}}} ∐_{j∈J} X(c_{j}) × S^{n−1} ⟶^{u} Z_{n−1} ×_{C} X ⟶^{f_{n−1}} Y_{n−1}

is G-homotopic to some cellular G-map v: ∐_{j∈J} X(c_{j})′ ×S^{n−1} → Y_{n−1}. Define Y_{n} by the G-pushout

\xymatrix{∐_{j∈J} X(c_{j})′ × S^{n−1} \ar[d]_{∐_{j∈J} id_{X(c_{j})′} ×i} \ar[r]^{v} & Y_{n−1} \ar[d] \\ ∐_{j∈J} X(c_{j})′ × D^{n} \ar[r] & Y_{n}}

This is a G-CW -complex whose dimension satisfies

dim(Y_{n}) ≤ max {n + sup { dim(X(c_{j})′) | j ∈ J}, dim(Y_{n−1})},

and the induction hypothesis applied to Y_{n−1} implies

dim(Y_{n}) ≤ n + sup { hdim^{G}(X(c)) | c ∈ ob(C)}.

Having shown this, we beg the reader’s pardon for omitting the actual construction of the G-homotopy equivalence f_{n}: Z_{n} ×_{C} X → Y_{n}. This construction is based on various standard cofibration arguments and is essentially the same as in the proof that the G-homotopy type of a G-CW -complex depends only on the G-homotopy classes of the attaching maps. (ii) There is an obvious natural map

f_{Z}: Z ×_{C} X^{H} → (Z ×_{C} X)^{H} ,

and analogously for Z_{n} in place of Z. We show by induction over n that the map f_{Z_{n}} is a homeomorphism. Then the claim follows for all finite-dimensional C-CW -complexes Z. The infinite-dimensional case follows using a colimit argument.

In the induction step, just note that the various homeomorphisms f_{Z_{n−1}} induce a map from the pushout

\xymatrix{∐_{j∈J} X(c_{j})^{H} × S^{n−1} \ar[d]_{∐_{j∈J} id_{X(c_{j})^{H}} ×i} \ar[r] & Z_{n−1} ×_{C} X^{H} \ar[d] \\ ∐_{j∈J} X(c_{j})^{H} × D^{n} \ar[r] & Z_{n} ×_{C} X^{H}}

to the pushout

\xymatrix{(∐_{j∈J} X(c_{j}) × S^{n−1})^{H} \ar[d]_{(∐_{j∈J} id_{X(c_{j})} ×i)^{H}} \ar[r] & (Z_{n−1} ×_{C} X)^{H} \ar[d] \\ (∐_{j∈J} X(c_{j}) × D^{n})^{H} \ar[r] & (Z_{n} ×_{C} X)^{H}}

and that a pushout of homeomorphisms is again a homeomorphism.

Lemma 4.5. Let {G_{i} | i ∈ I} be a directed system of groups. Let G = colim_{i∈I} G_{i} be its colimit. Let F be a family of subgroups of G such that every H ∈ F is contained in im(ψ_{i}) for some i ∈ I. We obtain a covariant functor G/ im(ψ_{?}): I → G-SPACESby sending i to G/ im(ψ_{i}).

Then EI ×_{I} G/ im(ψ_{?}) is G-homotopy equivalent to {•}.

Proof. We already know from Lemma 4.4 (i) that EI ×_{I} G/ im(ψ_{?}) has the G-homotopy type of a G-CW -complex (one can actually check easily that it is itself a G-CW -complex, the n-skeleton being EI_{n} ×_{I} G/ im(ψ_{?})). Hence, by the equivariant Whitehead theorem (see for instance [13, Theorem 2.4 on page 36]), it suffices to show that (EI ×_{I} G/ im(ψ_{?}))^{H} is weakly contractible for every subgroup H ⊆ G. Because of Lemma 4.4 (ii), it remains to show for every H ⊆ G that EI ×_{I} (G/ im(ψ_{?}))^{H} is weakly contractible.

Recall that EI ×_{I} (G/ im(ψ_{?}))^{H} is defined as ∐_{i∈I} EI(i) × (G/ im(ψ_{i}))^{H}/ ∼ for some tensor product-like equivalence relation ∼ (see [6, Definition 1.4]). Consider two points x_{1} and x_{2} in EI ×_{I} (G/ im(ψ_{?}))^{H}. Since G = ⋃_{i∈I} im(ψ_{i}) and, by assumption, I is directed and there is an index j ∈ I with H ⊆ im(ψ_{j}), we can find an index k ∈ I and z_{1}, z_{2} ∈ EI(k) such that (z_{l}, 1 · im(ψ_{k})) ∈ EI(k) × G/ im(ψ_{k}) lies in EI(k) × (G/ im(ψ_{k}))^{H} and represent x_{l} for l = 1, 2. As EI(k) is contractible and in particular path-connected, we can join z_{1} and z_{2} and hence x_{1} and x_{2} by a path. This shows that EI ×_{I} (G/ im(ψ_{?}))^{H} is path-connected.

It remains to show for every compact subspace C ⊆ EI ×_{I} (G/ im(ψ_{?}))^{H} that we can find a subspace D ⊆ EI ×_{I} (G/ im(ψ_{?}))^{H} such that C ⊆ D and such that D is homotopy equivalent to a discrete set. In the following, we use the functorial bar model as described in [6, page 230] as the model for EI. Notice that the n-cells in EI are indexed by the set of all sequences i_{0} < i_{1} < … < i_{n} of elements in I. Since C is compact, it is contained in the union of finitely many of the equivariant cells of EI ×_{I} G/ im(ψ_{?}). Such an equivariant cell is of the shape (mor_{I}(?, i) × D^{n}) ×_{I} G/ im(ψ_{?}) = G/ im(ψ_{i}) × D^{n} for some cell mor_{I}(?, i) × D^{n} of EI. Since I is directed, we can find an index j ∈ I such that C is already contained in (EI)_{≤j} ×_{I} G/ im(ψ_{?}), where (EI)_{≤j} is the I-CW -subcomplex of EI built of cells for which the indexing sequence i_{0} < i_{1} < … < i_{n} satisfies i_{n} ≤ j. Let k_{j}: I_{≤j} → I be the inclusion of the full subcategory I_{≤j} of I formed by the objects i ∈ I with i ≤ j. Then (EI)_{≤j} is isomorphic to the induction (k_{j})_{∗}E(I_{≤j}) of the bar model E(I_{≤j}) of the classifying space for the category I_{≤j}. However, I_{≤j} has a terminal object, namely j, so the canonical projection E(I_{≤j}) → mor_{I_{≤j}}(?, j) is a homotopy equivalence of contravariant I_{≤j}-spaces. It induces a homotopy equivalence

(EI)_{≤j} ×_{I} G/ im(ψ_{?}) ≅ (k_{j})_{∗}E(I_{≤j}) ×_{I} G/ im(ψ_{?})

⟶^{≃} (k_{j})_{∗} mor_{I_{≤j}}(?, j) ×_{I} G/ im(ψ_{?}) = G/ im(ψ_{j}).

Summarizing, C ⊆ D holds for D := (EI)_{≤j} ×_{I} G/ im(ψ_{?}), and D is homotopy equivalent to a discrete set.

Now we can give the proof of Theorem 4.3.

Proof. Fix a model E_{F}(G). We get a covariant functor G/ im(ψ_{?})×E_{F}(G): I → G-SPACES by sending i to G/ im(ψ_{i}) × E_{F}(G) and, for i ≤ j, by sending the morphism i → j in I to the G-map pr × id: G/ im(ψ_{i}) × E_{F}(G) → G/ im(ψ_{j}) × E_{F}(G), where pr is the obvious projection. From Lemma 4.5, we obtain a G-homotopy equivalence

EI ×_{I} (G/ im(ψ_{?}) × E_{F}(G)) ≅ (EI ×_{I} G/ im(ψ_{?})) × E_{F}(G)

⟶^{≃} {•} × E_{F}(G) = E_{F}(G).

The G-space G/ im(ψ_{i}) × E_{F}(G) is a G-CW -complex for every i and is Ghomeomorphic to G ×_{G_{i}} ψ_{i}^{∗}E_{F}(G), where ψ_{i}^{∗}E_{F}(G) is the G_{i}-CW -complex obtained from the G-CW -complex E_{F}(G) by restriction with ψ_{i}. One easily checks that ψ_{i}^{∗}E_{F}(G) is a model for E_{ψ_{i}^{∗}F}(G_{i}). Hence, the G-space G/ im(ψ_{i})×E_{F}(G) is G-homotopy equivalent to a G-CW -complex of dimension hdim^{G_{i}}(E_{ψ_{i}^{∗}F}(G_{i})). Now the claim follows from Lemma 4.4 (i) applied to Z = EI and X = G/ im(ψ_{?}) × E_{F}(G).

5 On the dimension of \underline{\underline{E}}G

In this section, we deal with the question what can be said about the dimension of G-CW -complexes which are models for \underline{\underline{E}}G. The same question for \underline{E}G has already been thoroughly investigated, and in many cases satisfying answers are known (see for instance [16], where further references to relevant papers are given).

5.1 Lower bounds for the homotopy dimension for VCY by the one for FIN

In this subsection, we give lower bounds for hdim^{G}(\underline{\underline{E}}G) in terms of hdim^{G}(\underline{E}G). We have introduced the notion of the homotopy dimension hdim^{G}(X) of a G-CW -complex X in Definition 4.1.

Proposition 5.1. Let F ⊆ G be families of subgroups of G.

(i) Let n ≥ 0 be an integer. Suppose that for any H ∈ G there is an ndimensional model for E_{F∩H}(H). Then

hdim^{G}(E_{F}(G)) ≤ n + hdim^{G}(E_{G}(G));

(ii) Suppose that there exists a finite H-CW -model for E_{F∩H}(H) for every H ∈ G and a finite G-CW -model for E_{G}(G). Then there exists a finite G-CW -model for E_{F}(G). The same is true if we replace “finite” by “of finite type” everywhere.

Proof. We only give the proof of assertion (i) since the one for assertion (ii) is similar.

Let Z be an m-dimensional G-CW -complex with isotropy groups in G. We will show that then Z × E_{F}(G) is G-homotopy equivalent to an (n + m)- dimensional G-CW -complex, which implies the claim of the proposition as E_{G}(G) × E_{F}(G) is a model for E_{F}(G).

We utilize induction over the dimension d of Z. If Z = ∅, then there is nothing to show, so let d ≥ 0. Crossing the G-pushout telling how Z_{d} arises from Z_{d−1} with E_{F}(G) yields a G-pushout

\xymatrix{∐_{i∈I_{d}} G/H_{i} × E_{F}(G) × S^{d−1} \ar[d] \ar[r]^{q} & Z_{d−1} × E_{F}(G) \ar[d] \\ ∐_{i∈I_{d}} G/H_{i} × E_{F}(G) × D^{d} \ar[r] & Z_{d} × E_{F}(G)}  (5.2)

There is a G-homotopy equivalence f_{i}: G ×_{H_{i}} E_{F∩H_{i}}(H_{i}) → G/H_{i} × E_{F}(G) since the restriction of E_{F}(G) to H_{i} is a H_{i}-CW -model for E_{F∩H_{i}}(H_{i}). We put f_{S} := ∐_{i} f_{i} × id_{S^{d−1}} and f_{D} := ∐_{i} f_{i} × id_{D^{d}}. Furthermore, by induction hypothesis, there is a G-homotopy equivalence h: Z′ → Z_{d−1} × E_{F}(G), where Z′ is an (n + d − 1)-dimensional G-CW -complex. Choose a G-homotopy inverse k of h.

For j: ∐_{i} G×_{H_{i}} E_{F∩H_{i}}(H_{i})×S^{d−1} → cyl(k ∘q ∘f_{S}) and p: cyl(k ∘q ∘f_{S}) → Z′ the obvious inclusion and projection, h ∘ p ∘ j and q ∘ f_{S} are G-homotopy equivalent. Since j is a G-cofibration, h∘p can be altered within its G-homotopy class to yield a G-map h′: cyl(k∘q∘f_{S}) → Z_{d−1}×E_{F}(G) such that h′∘j = q∘f_{S}. Now consider the (n + d)-dimensional G-CW -complex Z′′ which is defined by the G-pushout

\xymatrix{∐_{i∈I_{d}} G ×_{H_{i}} E_{F∩H_{i}}(H_{i}) × S^{d−1} \ar[d] \ar[r]^{j} & cyl(k ∘ q ∘ f_{S}) \ar[d] \\ ∐_{i∈I_{d}} G ×_{H_{i}} E_{F∩H_{i}}(H_{i}) × D^{d} \ar[r] & Z′′}  (5.3)

The G-homotopy equivalences f_{S}, f_{D} and h′ induce a map of G-pushouts from (5.3) to (5.2), and, as the left vertical arrows in these diagrams are Gcofibrations, Z_{d} × E_{F}(G) is G-homotopy equivalent to Z′′ by [13, Lemma 2.13].

Since any virtually cyclic group V admits a finite 1-dimensional model for \underline{E}V, Proposition 5.1 implies:

Corollary 5.4. Let G be a group.

(i) We have hdim^{G}(\underline{E}G) ≤ 1 + hdim^{G}(\underline{\underline{E}}G);

(ii) If there is a model for \underline{\underline{E}}G which is finite or of finite type, then there is a model for \underline{E}G which is finite or of finite type respectively.

Remark 5.5. The inequality appearing in Corollary 5.4 (i) is sharp (see Example 5.29, Theorem 5.33 (i) and Example 5.34).

Corollary 5.6. If G and H are groups, then

hdim^{G×H}(\underline{\underline{E}}(G × H)) ≤ hdim^{G}(\underline{\underline{E}}G) + hdim^{H}(\underline{\underline{E}}H) + 3.

Proof. Let F be the family of subgroups of G × H which are contained in subgroups of the shape V ×W for virtually cyclic subgroups V ⊆ G and W ⊆ H. Then \underline{\underline{E}}G × \underline{\underline{E}}H is a model for E_{F}(G × H). An element K in F contains ℤ^{n} for some n ∈ \{0, 1, 2\} as subgroup of finite index. This implies hdim^{K}(\underline{\underline{E}}K) ≤ 3 (see Example 5.21). Now apply Proposition 5.1 to G × H for VCY ⊆ F.

Remark 5.7. The inequality appearing in Corollary 5.6 is sharp, as can already be seen in the case G = H = ℤ (see Example 5.21).

For another example, let G_{−1} be the group appearing in Example 5.29. Then hdim^{G_{−1}}(\underline{\underline{E}}G_{−1}) = 3. Since the center of G_{−1} × G_{−1} is isomorphic to ℤ^{2} and vcd(G_{−1} × G_{−1}) = vcd(G_{−1}) + vcd(G_{−1}) = 8 by Lemma 5.14 (iv), Theorem 5.13 (iii) implies hdim^{G_{−1}×G_{1}} (\underline{\underline{E}}(G_{−1} × G_{−1})) = 9.

5.2 Upper bounds for the homotopy dimension for VCY by the one for FIN

The goal of this subsection is to give upper bounds for hdim^{G}(\underline{\underline{E}}G) in terms of hdim^{G}(\underline{E}G). Recall Notation 2.7.

Theorem 5.8. (i) Let G be a group satisfying M_{FIN ⊆VCY}. Suppose that one of the following conditions is satisfied:

(a) For every maximal infinite virtually cyclic subgroup V ⊆ G of type I

we have

dim(EW_{G}V) < ∞.

(b) We have: hdim^{G}(\underline{\underline{E}}G) < ∞.

Then:

hdim^{G}(\underline{\underline{E}}G) ≤ hdim^{G}(\underline{E}G) + 1;

(ii) Let G be a group satisfying NM_{FIN ⊆VCY}. Then:

hdim^{G}(\underline{\underline{E}}G) \begin{cases}= hdim^{G}(\underline{E}G) & if hdim^{G}(\underline{E}G) ≥ 2; \\ ≤ 2 & if hdim^{G}(\underline{E}G) ≤ 1. \end{cases}

Proof. (i) We first explain that condition (i)(b) implies condition (i)(a). Choose a finite-dimensional G-CW -model for \underline{\underline{E}}G. Let V ⊆ G be a maximal infinite virtually cyclic subgroup. The G-action on \underline{\underline{E}}G induces a N_{G}V -action by restriction and hence a W_{G}V -action on \underline{\underline{E}}G^{V}. Obviously \underline{\underline{E}}G^{V} is a W_{G}V -CW -complex which has finite dimension and is contractible (after forgetting the group action). It suffices to show that it is a model for EW_{G}V, i.e., that W_{G}V acts freely on \underline{\underline{E}}G^{V}. For x ∈ \underline{\underline{E}}G^{V} the isotropy group G_{x} under the G-action is a virtually cyclic subgroup of G which contains V. By the maximality of V we conclude G_{x} = V. Hence the isotropy group of x under the W_{G}V -action is trivial.

Hence it suffices to prove assertion (i) provided that condition (i)(a) is true. Let cyl(f_{V}) be the mapping cylinder of the N_{G}V -map f_{V} : \underline{E}N_{G}V → EW_{G}V

appearing in Corollary 2.10. Then we obtain from Corollary 2.10 a cellular Gpushout

\xymatrix{∐_{V ∈M} G ×_{N_{G}V} \underline{E}N_{G}V \ar[d]^{∐_{V ∈M} G×_{N_{G}V} i_{V}} \ar[r]^{i} & \underline{E}G \ar[d] \\ ∐_{V ∈M} G ×_{N_{G}V} cyl(f_{V}) \ar[r] & \underline{\underline{E}}G}

where the i_{V} are the obvious inclusions. Since res_{G}^{N_{G}V} \underline{E}G is an N_{G}V -CW - model for \underline{E}N_{G}V, it suffices to show

hdim^{W_{G}V} (EW_{G}V) ≤ hdim^{N_{G}V} (\underline{E}N_{G}V).  (5.9)

Let d = hdim^{W_{G}V} (EW_{G}V) if hdim^{W_{G}V} (EW_{G}V) is finite, or let d be any positive integer if hdim^{W_{G}V} (EW_{G}V) is infinite. We can choose a left ℤ[W_{G}V]- module M such that H_{W_{G}V}^{d} (EW_{G}V; M) ≠ 0 (see [4, Proposition 2.2 on page 185]). Since W_{G}V is torsionfree, V \\underline{E}N_{G}V is a free W_{G}V -CW -complex. By the equivariant Whitehead theorem (see [13, Theorem 2.4 on page 36]), the projection V \\underline{E}N_{G}V × EW_{G}V → V \\underline{E}N_{G}V is a W_{G}V -homotopy equivalence, where W_{G}V acts diagonally on the source. Since EW_{G}V is a free W_{G}V -CW -complex, we obtain a cohomology spectral sequence converging to H_{W_{G}V}^{p+q} (V \\underline{E}N_{G}V; M) whose E_{2}-term is given by

E_{2}^{p,q} = H_{W_{G}V}^{p} (EW_{G}V; H_{W_{G}V}^{q} (V \\underline{E}N_{G}V × W_{G}V; M)).

Here, W_{G}V acts diagonally from the left on V \\underline{E}N_{G}V × W_{G}V, and the W_{G}V - action on H_{W_{G}V}^{q} (V \\underline{E}N_{G}V × W_{G}V; M)) comes from the right W_{G}V -action on V \\underline{E}N_{G}V ×W_{G}V which is given by (x, w′)·w = (x, w′w) (and which commutes with the left W_{G}V -action). Now consider the homeomorphism

ϕ: V \\underline{E}N_{G}V × W_{G}V → V \\underline{E}N_{G}V × W_{G}V, (x, w) ↦ (wx, w).

It is equivariant with respect to the left W_{G}V -actions given by w · (x, w′) = (x, ww′) on the source and the diagonal action on the target. Moreover, it is equivariant with respect to the right W_{G}V -actions given by (x, w′) · w = (w^{−1}x, w′w) on the source and by (x, w′) · w = (x, w′w) on the target. Thus, we obtain an isomorphism of abelian groups

α: H_{W_{G}V}^{q} (V \\underline{E}N_{G}V × W_{G}V; M) ⟶^{H_{W_{G}V}^{q} (ϕ;M)} H_{W_{G}V}^{q} (V \\underline{E}N_{G}V × W_{G}V; M)

⟶^{μ} H^{q}(V \\underline{E}N_{G}V; M).

where μ is induced by the isomorphism of ℤ-cochain complexes

hom_{ℤ[W_{G}V]}(C_{∗}(V \\underline{E}N_{G}V × W_{G}V), M)

≅ hom_{ℤ[W_{G}V]}(C_{∗}(V \\underline{E}N_{G}V) ⊗_{ℤ} ℤ[W_{G}V], M) ≅ hom_{ℤ}(C_{∗}(V \\underline{E}N_{G}V); M).

The isomorphism α becomes an isomorphism of left ℤ[W_{G}V]-modules if we equip the target with the W_{G}V -action that comes from the following W_{G}V -action on the ℤ-chain complex hom_{ℤ}(C_{∗}(V \\underline{E}N_{G}V); M): namely, w ∈ W_{G}V acts by sending f to l_{w} ∘ f ∘ C_{∗}(l_{w^{−1}}), where l_{w} and l_{w^{−1}} denotes left multiplication with w and w^{−1}. The universal coefficient theorem for the ring ℤ yields an exact sequence of ℤ[W_{G}V]-modules

0 → Ext_{ℤ}^{1}(H_{q−1}(V \\underline{E}N_{G}V), M) → H^{q}(V \\underline{E}N_{G}V; M)

→ hom_{ℤ}(H_{q}(V \\underline{E}N_{G}V), M) → 0,

where the W_{G}V -actions on the first and third term are given by combining the obvious right W_{G}V -action on H_{∗}(V \\underline{E}N_{G}V) and the given left W_{G}V -action on M.

Now we separately treat the two cases that V is a virtually cyclic group of type I or of type II. In the first case, we can find an epimorphism α: V → ℤ with finite kernel. Then Eℤ, viewed as a V -space by restricting with α, is a model for \underline{E}V. Similarly, res_{N_{G}V}^{V} \underline{E}N_{G}V is a model for \underline{E}V. Hence, there is a V -homotopy equivalence \underline{E}N_{G}V → Eℤ, which induces a homotopy equivalence V \\underline{E}N_{G}V → ℤ\Eℤ. We conclude that H_{q}(V \\underline{E}N_{G}V) is infinite cyclic for q = 0, 1 and trivial for q ≥ 2. This implies that we obtain an isomorphism of ℤ[W_{G}V]-modules

H^{q}(V \\underline{E}N_{G}V; M) ⟶^{≅} hom_{ℤ}(H_{q}(V \\underline{E}N_{G}V), M).

Let ϵ: W_{G}V → {±1} the homomorphism which is uniquely determined by the property that w ∈ W_{G}V acts on H_{1}(V \\underline{E}N_{G}V) by multiplication with ϵ(w). Let M^{ϵ} be the ℤ[W_{G}V]-module whose underlying abelian group is M but on which the W_{G}V -action is twisted by ϵ, that is, w · x is defined to be ϵ(w)wx for w ∈ W_{G}V and x ∈ M. Then hom_{ℤ}(H_{1}(V \\underline{E}N_{G}V), M) is ℤ[W_{G}V]-isomorphic to M^{ϵ}. Summarizing, what we have shown so far is that E_{2}^{p,q} = H_{W_{G}V}^{p} (EW_{G}V; M^{ϵ}) for q = 1 and E_{2}^{p,q} = 0 for q ≥ 2. We get d < ∞ from condition (i)(a). Obviously E_{2}^{p,q} = 0 holds for p > d. So the E^{2}-term of the first quadrant spectral sequence looks like

⋮  ⋮  ⋮  ⋮  ⋮  ⋮ 0  ⋯  0  0  0  0  ⋯ 0  ⋯  0  0  0  0  ⋯ E_{2}^{0,1}  ⋯  E_{2}^{2,d−1}  H_{W_{G}V}^{d} (EW_{G}V; M^{ϵ})  0  0  ⋯ E_{2}^{0,0}  ⋯  E_{2}^{d−1,0}  E_{2}^{d,0}  0  0  ⋯

Hence we obtain an isomorphism

H_{W_{G}V}^{d+1} (V \\underline{E}N_{G}V; M) ≅ H_{W_{G}V}^{d} (EW_{G}V; M^{ϵ}).

Since (M^{ϵ})^{ϵ} = M and H_{W_{G}V}^{d} (EW_{G}V; M) ≠ 0, this implies that we have H_{W_{G}V}^{d+1} (V \\underline{E}N_{G}V; M^{ϵ}) ≠ 0. It follows that hdim^{N_{G}V} (\underline{E}N_{G}V) ≥ d + 1, and hence assertion (5.9) is true in the case that V is of type I.

It remains to treat the (easier) case II, in which there is an epimorphism α: V → D_{∞} onto the infinite dihedral group. Just as above, we obtain a homotopy equivalence V \\underline{E}N_{G}V → D_{∞}\\underline{E}D_{∞}, and as D_{∞}\\underline{E}D_{∞} is contractible, H_{q}(V \\underline{E}N_{G}V) is trivial for q ≥ 1 and is isomorphic as a ℤ[W_{G}V]-module to ℤ with the trivial W_{G}V -action for q = 0. Hence, the cohomology spectral sequence yields an isomorphism

H_{W_{G}V}^{d} (V \\underline{E}N_{G}V; M) ≅ H_{W_{G}V}^{d} (EW_{G}V; M),

whence it follows that H_{W_{G}V}^{d} (V \\underline{E}N_{G}V; M) ≠ 0. So hdim^{N_{G}V} (\underline{E}N_{G}V) ≥ d, and we get assertion (5.9). (ii) From Corollary 2.11 we get a cellular G-pushout

\xymatrix{∐_{V ∈M} G ×_{V} \underline{E}V \ar[d]^{∐_{V ∈M} G×_{V} i_{V}} \ar[r]^{i} & \underline{E}G \ar[d] \\ ∐_{V ∈M} G ×_{V} cone(\underline{E}V) \ar[r] & \underline{\underline{E}}G}

where M is a complete system of representatives of the conjugacy classes of maximal infinite virtually cyclic subgroups and i_{V} is the obvious inclusion. Since each V is virtually cyclic, we can choose a 1-dimensional model for \underline{E}V. Then cone(\underline{E}V) is 2-dimensional. This implies that there exists a max\{d, 2\}-model for \underline{\underline{E}}G if there is a d-dimensional model for \underline{E}G. Hence it remains to prove hdim^{G}(\underline{E}G) ≤ hdim^{G}(\underline{\underline{E}}G) provided that hdim^{G}(\underline{E}G) ≥ 2 holds.

Put d = hdim^{G}(\underline{E}G). We get from [13, Proposition 11.10 on page 221] a contravariant ℤOr_{FIN} (G)-module M with H^{d}(hom_{ℤOr _{FIN} (G)}(C_{∗}(\underline{E}G), M)) ≠ 0. Define a contravariant ℤOr_{VCY}(G)-module \overline{M} by putting \overline{M}(G/H) = M(G/H) if H ∈ FIN and \overline{M}(G/H) = 0 if H ∉ FIN. If f: G/H → G/K is a morphism in Or_{VCY}(G), define \overline{M}(f) = M(f) if H, K ∈ FIN, and \overline{M}(f) = 0 otherwise. If j: Or_{FIN} (G) → Or_{VCY}(G) is the inclusion, then the restriction j^{∗}\overline{M} agrees with M. The adjunction between induction and restriction yields an isomorphism of ℤ-chain complexes

hom_{ℤOr _{FIN} (G)}(C_{∗}(\underline{E}G), M)) ≅ hom_{ℤOr _{FIN} (G)}(C_{∗}(\underline{E}G), j^{∗}\overline{M}))

≅ hom_{ℤOr _{VCY} (G)}(j_{∗}C_{∗}(\underline{E}G), \overline{M})).

Since there exists a 1-dimensional model for \underline{E}V if V ∈ M and since d ≥ 2, we obtain an epimorphism

H^{d}(hom_{ℤOr _{VCY} (G)}(C_{∗}(\underline{\underline{E}}G), \overline{M})) → H^{d}(hom_{ℤOr _{VCY} (G)}(j_{∗}C_{∗}(\underline{E}G), \overline{M}))

from the long exact cohomology sequence of the pair (\underline{\underline{E}}G, \underline{E}G) with coefficients in \overline{M}. This implies H^{d}(hom_{ℤOr _{VCY} (G)}(C_{∗}(\underline{\underline{E}}G), \overline{M})) ≠ 0 and hence d ≤ hdim^{G}(\underline{\underline{E}}G). This finishes the proof of Theorem 5.8.

Remark 5.10. Condition (i)(a) appearing Theorem 5.8 could be avoided if one could show the following: If 1 → C → Γ → Q → 1 is an extension of groups such that C is infinite cyclic, Q is torsionfree and there exists a finitedimensional model for BΓ, then there exists a finite-dimensional model for BQ. However, it is unlikely that such a statement is true, although we have no explicit counterexample.

Suppose that we have such an extension for which no finite-dimensional model for BQ exists. Then the Gysin sequence, whose existence follows from spectral sequence argument appearing in Theorem 5.8, implies that for every ℤQ-module M and n ≥ dim(BΓ) we have H_{Q}^{n}(EQ; M) ≅ H_{Q}^{n+2}(EQ; M), i.e., the cohomology of Q is for large n two-periodic with respect to any ℤQcoefficient module M.

Remark 5.11. The inequality appearing in Theorem 5.8 (i) is sharp. For instance, we have hdim^{ℤ^{n}}(\underline{E}ℤ^{n}) = n and hdim^{ℤ^{n}}(\underline{\underline{E}}ℤ^{n}) = n + 1 for n ≥ 2 (see Example 5.21).

Example 5.12 (Counterexample to a conjecture of Connolly-Fehr-mann-Hartglass). Let G be the fundamental group of a closed hyperbolic manifold M of dimension d. Then G is torsionfree and satisfies NM_{FIN ⊆VCY} by Example 3.6. This implies, using Theorem 5.8 (ii)

d = hdim^{\{1\}}(M) = hdim^{G}(EG) = hdim^{G}(\underline{\underline{E}}G).

Obviously, G is a discrete cocompact subgroup of the isometry group of the d-dimensional hyperbolic space, which is a Lie group with only finitely many path components.

Thus, we get counterexamples to a conjecture stated by Connolly-Fehrmann-Hartglass [5, Conjecture 8.1], which predicts that hdim^{G}(\underline{\underline{E}}G) = d + 1.

5.3 Virtually poly-ℤ groups

In this subsection, we give a complete calculation of hdim^{G}(\underline{\underline{E}}G) for virtually poly-ℤ groups. Surprisingly, it will depend not only on vcd(G) = hdim^{G}(\underline{E}G).

A group G is poly-ℤ if there is a finite sequence {1} = G_{0} ⊆ G_{1} ⊆ … ⊆ G_{n} = G of subgroups such that G_{i−1} is normal in G_{i} with infinite cyclic quotient G_{i}/G_{i−1} for i = 1, 2, …, n.

We have introduced notations like [VCY \ FIN], [V] and N_{G}[V] at the beginning of Section 2. In particular, recall Definition 2.2.

Theorem 5.13. Let G be a virtually poly ℤ group which is not virtually cyclic. Let [VCY \ FIN]_{f} be the subset of [VCY \ FIN] consisting of those elements [V] for which N_{G}[V] has finite index in G. Let [VCY \ FIN]_{f}/G be its quotient under the G-action coming from conjugation. Then

hdim^{G}(\underline{E}G) = vcd(G),

and precisely one of the following cases occurs:

(i) The set [VCY \ FIN]_{f}/G is empty. This is equivalent to the condition that every finite index subgroup of G has a finite center. In this case,

hdim^{G}(\underline{\underline{E}}G) = vcd(G);

(ii) The set [VCY \ FIN]_{f}/G contains exactly one element [V] · G. This is equivalent to the condition that there exists an infinite normal cyclic subgroup C ⊆ G and for every infinite cyclic subgroup D ⊆ G with [G : N_{G}D] < ∞ we have C ∩ D ≠ {1}. Then there is the following dichotomy:

(a) For every [W] · G ∈ [VCY \ FIN]/G such that [W] · G ≠ [V] · G we have vcd(N_{G}[W]) ≤ vcd(G) − 2. In this case, vcd(G) ≥ 4 and

hdim^{G}(\underline{\underline{E}}G) = vcd(G) − 1;

(b) There exists [W] · G ∈ [VCY \ FIN]/G such that [W] · G ≠ [V] · G and vcd(N_{G}[W]) = vcd(G) − 1. In this case, vcd(G) ≥ 3 and

hdim^{G}(\underline{\underline{E}}G) = vcd(G);

(iii) The set [VCY \FIN]_{f}/G contains more than one element. This is equivalent to the condition that there is a finite index subgroup of G whose center contains a subgroup isomorphic to ℤ^{2}. In this case,

hdim^{G}(\underline{\underline{E}}G) = vcd(G) + 1.

The proof of Theorem 5.13 needs some preparations.

Lemma 5.14. (i) Subgroups and quotients of poly-ℤ groups are again poly-ℤ. If 1 → G_{0} → G_{1} → G_{2} → 1 is an extension of groups and G_{0} and G_{2} are poly-ℤ groups, then G is a poly-ℤ group.

The same statements are true if one replaces “poly-ℤ” by “virtually poly-ℤ” everywhere;

(ii) Let G be a poly-ℤ group of cohomological dimension cd(G) = d. Then H^{d}(G) is isomorphic to ℤ or to ℤ/2;

(iii) Let G be a poly-ℤ group. Suppose that there is a finite sequence {1} = G_{0} ⊆ G_{1} ⊆ … ⊆ G_{n} = G of subgroups such that G_{i−1} is normal in G_{i} with infinite cyclic quotient G_{i}/G_{i−1} for i = 1, 2, …, n. Then

n = cd(G);

(iv) If 1 → G_{0} → G_{1} → G_{2} → 1 is an extension of virtually poly-ℤ groups,

then

vcd(G_{1}) = vcd(G_{0}) + vcd(G_{2});

(v) A virtually poly-ℤ group G has the following property:

(A) If ϕ: G → G is an automorphism, g ∈ G is an element of infinite order and a, b ∈ ℤ are integers such that ϕ(g^{a}) = g^{b}, then a = ±b.

Proof. (i) The elementary proof of this assertion is left to the reader. (ii) We use induction over the number n for which there is a finite sequence {1} = G_{0} ⊆ G_{1} ⊆ … ⊆ G_{n} = G of subgroups such that G_{i−1} is normal in G_{i} with infinite cyclic quotient G_{i}/G_{i−1} for i = 1, 2, …, n. If n = 0, then G is trivial and the claim is obviously true. The induction step from n − 1 to n ≥ 1 is done as follows.

By the induction hypothesis, the claim is true for G_{n−1}. This means that cd(G_{n−1}) = n − 1, that H^{n−1}(G_{n−1}) is isomorphic to ℤ or ℤ/2 and that H^{i}(G_{n−1}) = 0 for i ≥ n. We have the extension 1 → G_{n−1} → G → ℤ → 1 if we identify G/G_{n−1} with ℤ. Let f: G_{n−1} → G_{n} be the group automorphism induced by conjugation with some preimage in G of a generator of ℤ. The associated Hochschild-Serre cohomology spectral sequence yields the exact Wang sequence

H^{n−1}(G_{n−1}) ⟶^{id −H^{n−1}(f)} H^{n−1}(G_{n−1}) → H^{n}(G) → H^{n}(G_{n−1}) = 0.

Since H^{n−1}(G_{n−1}) is isomorphic to ℤ or ℤ/2, the automorphism H^{n−1}(f) is ± id. Hence H^{n}(G) ≅ ℤ if and only if H^{n−1}(G_{n−1}) ≅ ℤ and H^{n−1}(f) = id, and H^{n}(G) ≅ ℤ/2 if and only if either H^{n−1}(G_{n−1}) ≅ ℤ and H^{n−1}(f) = − id or H^{n−1}(G_{n−1}) ≅ ℤ/2.

(iii) We use induction over the number n as above. Since H^{n}(G) is non-trivial, we have cd(G) ≥ n. On the other hand, the cohomological dimension is subadditive under extensions, i.e., cd(G) ≤ cd(G_{n−1}) + cd(ℤ) = (n − 1) + 1 = n. We conclude cd(G) = n. (iv) One easily constructs an exact sequence of poly-ℤ groups 1 → G′_{0} → G′_{1} → G′_{2} → 1 such that G′_{i} is a subgroup of finite index in G_{i} for i = 0, 1, 2. Now, it suffices to show that cd(G′_{1}) = cd(G′_{0}) + cd(G′_{2}). By induction over the number n as above but with G′_{2} in place of G, the claim reduces to the case that G′_{2} = ℤ, which follows from assertion (iii).

(v) Consider a group extension 1 → G_{0} ⟶^{i} G_{1} ⟶^{p} G_{2} → 1 such that G_{0} is characteristic in G_{1} and G_{0} and G_{2} have property (A). Then G_{1} has property (A) by the following argument. Consider an automorphism ϕ_{1} : G_{1} → G_{1}. It induces automorphisms ϕ_{0} : G_{0} → G_{0} and ϕ_{2} : G_{2} → G_{2} satisfying ϕ_{1} ∘i = i∘ϕ_{0} and ϕ_{2} ∘ p = p ∘ ϕ_{1} since G_{0} is characteristic by assumption. Let g ∈ G be of infinite order and a, b ∈ ℤ such that ϕ_{1}(g^{a}) = g^{b}. Then ϕ_{2}(p(g)^{a}) = p(g)^{b}, and if p(g) has infinite order, we see that a = ±b since G_{2} has property (A) by assumption. If p(g) has finite order, we can choose c ∈ ℤ, c ≠ 0 such that g^{c} ∈ G_{0}. Then ϕ_{1}(g^{a}) = g^{b} implies ϕ_{0}((g^{c})^{a}) = (g^{c})^{b} in G_{0} and hence a = ±b since G_{0} has property (A) by assumption. Hence, G_{1} has property (A).

Now we prove by induction over vcd(G) that a virtually poly-ℤ group G has property (A). The induction beginning vcd(G) = 0 is trivial since then G is finite. The induction step is done as follows. Let K ⊆ G be a subgroup of finite index such that K is poly-ℤ. Then K_{0} = ⋂{ϕ(K) | ϕ ∈ aut(G)} is a characteristic subgroup of G. Since G is finitely generated and, thus, contains only finitely many subgroups of any given index, the intersection defining K_{0} is an intersection of finitely many groups of finite index. Hence, K_{0} is a characteristic subgroup of finite index in G. Since every finite group has property (A), the group G has property (A) if K_{0} has. Consider the group extension 1 → K_{1} → K_{0} → H_{1}(K_{0})/ tors(H_{1}(K_{0})) → 1. Obviously, K_{1} is a characteristic subgroup of K_{0}. Since K_{0}, being a subgroup of the poly-ℤ group K, is poly-ℤ, the abelian group H_{1}(K_{0})/ tors(H_{1}(K_{0}) is isomorphic to ℤ^{n} for some n ≥ 1. This implies that K_{1} is a poly-ℤ group satisfying vcd(K_{1}) < vcd(K_{0}) = vcd(G). By induction hypothesis, K_{1} has property (A). Clearly, ℤ^{n} also has property (A). This implies that K_{0} and hence G have property (A). This finishes the proof of Lemma 5.14.

The statements about the top homology of a poly-ℤ group G appearing in Lemma 5.14 would be obvious if there were closed manifold models for BG. Instead of dealing with this difficult question we have preferred to give an easy homological argument.

Lemma 5.15. Let G be a virtually poly-ℤ group. Let V ⊆ G be an infinite virtually cyclic group. Then we can find an infinite cyclic subgroup C ⊆ V such that [V] = [C] ∈ [VCY \ FIN] and N_{G}C = N_{G}[V] holds, and such that a model for E_{VCY[V]}(N_{G}[V]) is given by \underline{E}W_{G}C considered as an N_{G}[V] = N_{G}C-CW - complex by restriction with the canonical projection N_{G}C → W_{G}C.

Proof. We can assume that V itself is infinite cyclic (otherwise, we pass to a subgroup of finite index). Since N_{G}[V] is a subgroup of the virtually poly-ℤ group G, it is virtually poly-ℤ and, in particular, finitely generated. Let {g_{1}, g_{2}, …, g_{r}} be a finite generating set. Fix a generator c ∈ V. Since g_{i} ∈ N_{G}[V], we can find integers a_{i}, b_{i} ∈ ℤ different from zero such that g_{i}c^{a_{i}}g_{i}^{−1} = c^{b_{i}}. Property (A) (see Lemma 5.14 (v)) implies b_{i} = ±a_{i}. Put a = a_{1}a_{2} ⋯ a_{r}. Then g_{i}c^{a}g_{i}^{−1} = c^{±a} holds for i = 1, 2, …, r. Let C be the infinite cyclic subgroup of V generated by c^{a}. Then, of course, [C] = [V]. Furthermore, g_{i} ∈ N_{G}C for i = 1, 2, …, r, which implies N_{G}[V] ⊆ N_{G}C. Hence, N_{G}[V] = N_{G}C.

It remains to prove that \underline{E}W_{G}C is a model for E_{VCY[C]}(N_{G}C). This follows from the fact that, under the projection N_{G}C → W_{G}C, the family of all virtually cyclic subgroups of N_{G}C whose intersection with C is infinite coincides with the family of all finite subgroups of W_{G}C.

Now we are ready to prove Theorem 5.13.

Proof. Because of Lemma 5.15, we can choose a set I of infinite cyclic subgroups of G and models for \underline{E}N_{G}C and \underline{E}W_{G}C for C ∈ I such that the following properties hold:

• For every element η ∈ [VCY \ FIN]/G, there exists precisely one C ∈ I such that [C] · G = η;

• N_{G}[C] = N_{G}C for all C ∈ I;

• For all C ∈ I, a model for E_{VCY[C]}(N_{G}[C]) is given by \underline{E}W_{G}C, considered as an N_{G}C-CW -complex by restriction with the canonical projection N_{G}C → W_{G}C.

From Theorem 2.3 we obtain a G-pushout

\xymatrix{∐_{C∈I} G ×_{N_{G}C} \underline{E}N_{G}C \ar[d]^{∐_{C∈I} id_{G} ×_{N_{G}C}f_{C}} \ar[r]^{i} & \underline{E}G \ar[d] \\ ∐_{C∈I} G ×_{N_{G}C} \underline{E}W_{G}C \ar[r] & \underline{\underline{E}}G}  (5.16)

such that f_{C} is a cellular N_{G}C-map for every C ∈ I and i is an inclusion of G-CW -complexes.

In the remainder of the proof, we will frequently use that Lemma 5.14 (iv) implies

vcd(W_{G}C) = vcd(N_{G}C) − 1;
vcd(N_{G}C) ≤ vcd(G);
vcd(N_{G}C) = vcd(G) ⇔ [G : N_{G}C] < ∞.

Furthermore, we conclude from [16, Example 5.26 on page 305] that

hdim^{G}(\underline{E}G) = vcd(G);
hdim^{G}(\underline{E}N_{G}C) = vcd(N_{G}C)  for C ∈ I;
hdim^{G}(\underline{E}W_{G}C) = vcd(N_{G}C) − 1  for C ∈ I.

(i) As [VCY \FIN]_{f}/G = ∅, we must have hdim^{G}(\underline{E}W_{G}C) < hdim^{G}(\underline{E}N_{G}C) < vcd(G), whereas hdim^{G}(\underline{E}G) = vcd(G). Hence, we conclude from the Gpushout (5.16) together with Remark 2.5 that hdim^{G}(\underline{\underline{E}}G) ≤ vcd(G). In order to show hdim^{G}(\underline{\underline{E}}G) ≥ vcd(G) we can assume without loss of generality that G is poly-ℤ since there is a poly-ℤ subgroup G′ of finite index and the restriction of \underline{\underline{E}}G to G′ is a model for \underline{\underline{E}}G′. The G-pushout (5.16) yields a pushout of CW -complexes

\xymatrix{∐_{C∈I} N_{G}C\\underline{E}N_{G}C \ar[d]^{∐_{C∈I} N_{G}C\f_{C}} \ar[r]^{i} & G\\underline{E}G \ar[d] \\ ∐_{C∈I} W_{G}C\\underline{E}W_{G}C \ar[r] & G\\underline{\underline{E}}G}  (5.17)

From the associated long exact Mayer-Vietoris sequence, we obtain an epimorphism H^{vcd(G)}(G\\underline{\underline{E}}G) → H^{vcd(G)}(G\\underline{E}G). However, H^{vcd(G)}(G\\underline{E}G) is nontrivial by Lemma 5.14 (ii), so H^{vcd(G)}(G\\underline{\underline{E}}G) is also non-trivial. This means that hdim^{G}(\underline{\underline{E}}G) ≥ vcd(G).

As C_{G}C has finite index in N_{G}C for C ∈ I, the condition [VCY\FIN]_{f}/G = ∅ is equivalent to the condition that for every infinite cyclic subgroup C ⊆ G we have [G : C_{G}C] = ∞. Obviously, this is true if and only if the center of any finite index subgroup of G is finite. This finishes the proof of assertion (i). (iii) We conclude from the G-pushout (5.16) together with Remark 2.5 that hdim^{G}(\underline{\underline{E}}G) ≤ vcd(G) + 1 as hdim^{G}(\underline{E}N_{G}C), hdim^{G}(\underline{E}W_{G}C) and hdim^{G}(\underline{E}G) are less or equal to vcd(G). It remains to show that hdim^{G}(\underline{\underline{E}}G) ≥ vcd(G) + 1, and it suffices to do this for a subgroup of finite index. Hence, we can assume without loss of generality that G is poly-ℤ. We obtain from the pushout (5.17) the short exact sequence

H^{vcd(G)}(G\\underline{E}G) → ∏_{C∈I} H^{vcd(G)}(N_{G}C\\underline{E}N_{G}C) → H^{vcd(G)+1}(G\\underline{\underline{E}}G) → 0.

If we can show that H^{vcd(G)}(G\\underline{E}G) → ∏_{C∈I} H^{vcd(G)}(N_{G}C\\underline{E}N_{G}C) is not surjective, then H^{vcd(G)+1}(G\\underline{\underline{E}}G) is non-trivial, implying hdim^{G}(\underline{\underline{E}}G) ≥ vcd(G)+ 1. By Lemma 5.14 (ii), H^{vcd(G)}(G\\underline{E}G) and H^{vcd(N_{G}C)}(N_{G}C\\underline{E}N_{G}C) are nontrivial cyclic groups. Thus, H^{vcd(G)}(G\\underline{E}G) → ∏_{C∈I} H^{vcd(G)}(N_{G}C\\underline{E}N_{G}C) cannot be surjective if there is more than one element C ∈ I with vcd(N_{G}C) = vcd(G), which is indeed the case here since, by assumption, [VCY \ FIN]_{f}/G contains more than one element.

We still want to show that this latter condition is equivalent to the condition that there is a finite index subgroup of G whose center contains a copy of ℤ^{2}.

Suppose first that [VCY \ FIN]_{f}/G contains two different elements [V_{1}] · G and [V_{2}] · G. Lemma 5.15 shows that we may assume that V_{i} is infinite cyclic and N_{G}V_{i} = N_{G}[V_{i}] for i = 1, 2. Then, as C_{G}V_{i} has finite index in N_{G}V_{i} for i = 1, 2, the subgroup C_{G}V_{1} ∩ C_{G}V_{2} ⊆ G is also of finite index. Choose infinite cyclic groups C_{1} and C_{2} such that C_{1} ⊆ V_{1}∩C_{G}V_{2} and C_{2} ⊆ V_{2}∩C_{G}V_{1}, and let H ⊆ G be the subgroup generated by C_{1} and C_{2}. Then C_{G}H contains C_{G}V_{1} ∩ C_{G}V_{2}. Hence C_{G}H has finite index in G and H ⊆ C_{G}H. In particular, H is abelian. Since [V_{1}] ≠ [V_{2}], we have C_{1} ∩ C_{2} = {1}. So H is isomorphic to ℤ^{2}, and we have found a finite index subgroup of G, namely C_{G}H, whose center contains ℤ^{2}.

Conversely, suppose that G′ ⊆ G is a subgroup of finite index whose center contains ℤ^{2}. By passing to a subgroup of finite index in G′, we can arrange for G′ ⊆ G to be torsionfree and normal with finite quotient G/G′. Since cent(G′) is isomorphic to ℤ^{n} for some n ≥ 2, it contains infinitely many maximal infinite cyclic subgroups. As G/G′ is finite, [VCY \ FIN]_{f}/G must be infinite and, in particular, contains more than one element. (ii) We claim that [VCY \FIN]_{f}/G consists of precisely one element if and only if G contains an infinite cyclic normal subgroup C and for every infinite cyclic subgroup D ⊆ G with [G : N_{G}D] < ∞ we have C∩D ≠ {1}. The “if”-statement is obvious, so let us show that G contains an infinite cyclic normal subgroup if [VCY \ FIN]_{f}/G consists of precisely one element [V] · G. Again, just as in the proof of (iii), we can assume by Lemma 5.15 that V is infinite cyclic and [G : C_{G}V] < ∞. This implies that K := ⋂_{g∈G} gC_{G}V g^{−1} is a normal subgroup of G of finite index. Thus, the center cent(K) of K, being a characteristic subgroup of K, is also normal in G. Moreover, we conclude from assertion (iii) that cent(K) does not contain ℤ^{2} as subgroup of finite index. Hence, it is isomorphic to ℤ × A for some finite abelian subgroup A. If a is the order of A, then C := {a · x | x ∈ cent(K)} is a characteristic infinite cyclic subgroup of cent(K) and hence normal in G. The claim follows.

In the remainder, let C_{0} ⊆ G be an infinite cyclic normal subgroup. This means that [VCY \ FIN]_{f}/G = {[C_{0}] · G}. Since we assume that G is not virtually cyclic, vcd(G) ≥ 2. In fact, we have vcd(G) > 2. Namely, if vcd(G) = 2, then G contains a subgroup of finite index isomorphic to ℤ^{2}, which is only possible in the case (iii) and not in the case (ii) under consideration.

Next, we want to show that vcd(G) ≥ 4 in the case (ii)(a) and vcd(G) ≥ 3 in the case (ii)(b). We can assume that C_{0} ⊆ G is central and that G is poly-ℤ for, otherwise, we can replace G by a subgroup of finite index. Choose an element g ∈ G whose image under the projection G → G/C_{0} is an element of infinite order. Let D be the cyclic subgroup generated by g ∈ G. Then D ∩ C_{0} = {1} and N_{G}D contains a subgroup isomorphic to ℤ^{2}. Hence vcd(N_{G}D) ≥ 2 and [D] ≠ [C_{0}]. The assumptions imply vcd(D) ≤ vcd(G) − 2, which in turn implies vcd(G) ≥ 4 in the case (ii)(a) and vcd(G) ≥ 3 in the case (ii)(b).

In order to get a model for \underline{\underline{E}}G of the desired dimension, we have to investigate the G-pushout (5.16) more closely and use some machinery. Let \underline{\underline{E}}G be the G-CW -model obtained from the G-pushout (5.16). Given a natural number d ≥ 3, we will show that there exists a G-CW -complex of dimension d which is G-homotopy equivalent to \underline{\underline{E}}G by checking that for every contravariant ℤOr(G)-module M the Bredon cohomology H_{ℤOr (G)}^{d+1}(\underline{\underline{E}}G; M) vanishes and for every i ≥ d + 1 the homology of \underline{\underline{E}}G^{H} vanishes for H ⊆ G (see [13, Proposition 11.10 on page 221 and Theorem 13.19 on page 268]). The latter condition is obviously satisfied since \underline{\underline{E}}G^{H} is contractible or empty. It remains to prove H_{ℤOr (G)}^{d+1}(\underline{\underline{E}}G; M) = 0.

Let C ∈ I be the unique element for which vcd(N_{G}C) = vcd(G). After replacing C by C_{0}, we can actually assume that N_{G}C = G. Let us put d = vcd(G) if vcd(N_{G}D) = vcd(G) − 1 for at least one D ∈ I with D ≠ C and d = vcd(G) − 1 if vcd(N_{G}D) ≤ vcd(G) − 2 holds for all D ∈ I with D ≠ C. Then we have d ≥ 3 and, in all cases, vcd(N_{G}D) < d for all D ∈ I with D ≠ C. In particular, H_{ℤOr (G)}^{k}(G×_{N_{G}D} \underline{E}N_{G}D; M) as well as H_{ℤOr (G)}^{k}(G×_{N_{G}D} \underline{E}W_{G}D; M) vanish for all D ∈ I with D ≠ C if k ≥ d. The long exact Mayer-Vietoris sequence for the Bredon cohomology associated to the G-pushout (5.16) yields the exact sequence

H_{ℤOr (G)}^{d}(\underline{E}G; M) ⟶^{id} H_{ℤOr (G)}^{d}(\underline{E}G; M) → H_{ℤOr (G)}^{d+1}(\underline{\underline{E}}G; M)

→ H_{ℤOr (G)}^{d+1}(\underline{E}G; M) ⟶^{id} H_{ℤOr (G)}^{d+1}(\underline{E}G; M).

Hence, H_{ℤOr (G)}^{d+1}(\underline{\underline{E}}G; M) is trivial and, as explained above, we can find a ddimensional G-CW -model for \underline{\underline{E}}G. We conclude that hdim^{G}(\underline{\underline{E}}G) ≤ vcd(G)−1 in the case (ii)(a) and hdim^{G}(\underline{\underline{E}}G) ≤ vcd(G) in the case (ii)(b).

Finally, we prove hdim^{G}(\underline{\underline{E}}G) ≥ vcd(G) − 1 in the case (ii)(a) as well as hdim^{G}(\underline{\underline{E}}G) ≥ vcd(G) in the case (ii)(b). The first inequality follows from Corollary 5.4. Concerning the second, we can assume that G is poly-ℤ. Consider the long exact Mayer-Vietoris sequence associated to the pushout (5.17). Since a poly-ℤ-group is torsionfree, N_{G}C = G and cd(EW_{G}D) ≤ d−2 for D ∈ I with D ≠ C, it yields the long exact sequence

H^{d−1}(G) ⟶^{id ×j} H^{d−1}(G) × ∏_{D∈I,D≠C} H^{d−1}(N_{G}D) → H^{d}(G\\underline{\underline{E}}G)

→ H^{d}(G) ⟶^{id} H^{d}(G).

As H^{d−1}(N_{G}D) is non-trivial if D ∈ I is such that vcd(N_{G}D) = d − 1 (see Lemma 5.14 (ii)), we conclude H^{d}(G\\underline{\underline{E}}G) ≠ 0 and hence hdim^{G}(\underline{\underline{E}}G) ≥ d = vcd(G). This finishes the proof of Theorem 5.13.

Corollary 5.18. Let G be a virtually poly-ℤ group. Suppose that the group G′ is a subgroup of G of finite index or that G′ is the quotient of G by a finite normal subgroup. Then

hdim^{G′}(\underline{E}G′) = hdim^{G}(\underline{E}G);
hdim^{G′}(\underline{\underline{E}}G′) = hdim^{G}(\underline{\underline{E}}G).

Proof. We get vcd(G′) = vcd(G) from Lemma 5.14 (iv). Hence, hdim^{G′}(\underline{E}G′) = hdim^{G}(\underline{E}G) follows immediately from Theorem 5.13.

The second assertion of the corollary will follow from assertions (i), (ii) and (iii) of Theorem 5.13. We only need to show that every subgroup of G′ of finite index has a finite center if and only if every subgroup of G of finite index has a finite center and that G′ contains a finite index subgroup whose center contains ℤ^{2} if and only if G contains a finite index subgroup whose center contains ℤ^{2}.

We begin with the case that G′ is a subgroup of finite index in G. A finitely generated abelian group A satisfies vcd(A) ≥ 2 if and only if it contains a subgroup isomorphic to ℤ^{2}. Obviously, a subgroup H ⊆ G satisfies [G : H] < ∞ if and only if [G′ : (G′ ∩ H)] < ∞. Now the claims follow from the observation that for a subgroup H ⊆ G we have cent(H)∩G′ ⊆ cent(H ∩G′) and [cent(H) : (cent(H) ∩ G′)] < ∞, which implies that vcd(cent(H)) ≤ vcd(cent(H ∩ G′)) holds.

It remains to treat the case that there is an extension 1 → F → G ⟶^{p} G′ → 1 in which the group F is finite. Let K ⊆ G be a subgroup, then p(cent(K)) ⊆ cent(p(K)), which implies

vcd(cent(K)) ≤ vcd(cent(p(K))).  (5.19)

Let H ⊆ G′ be a subgroup. Since G is virtually poly-ℤ, by Lemma 5.14 (i) the same is true for the subgroup p^{−1}(H). In particular, p^{−1}(H) is finitely generated, say by the finite set {s_{1}, s_{2}, …, s_{r}}. Consider the map

γ: p^{−1}(cent(H)) → ∏_{i=1}^{r} F, h ↦ {[h, s_{i}] | i = 1, 2, … r}.

We have p(cent(p^{−1}(H)) ⊆ cent(p(p^{−1}(H)) = cent(H) and hence cent(p^{−1}(H)) is a subgroup of p^{−1}(cent(H)). Now one easily checks that γ induces an injection p^{−1}(cent(H))/ cent(p^{−1}(H)) → ∏_{i=1}^{r} F, which means that cent(p^{−1}(H)) has finite index in p^{−1}(cent(H)). As F is finite, we conclude that

vcd(cent(p^{−1}(H))) = vcd(cent(H)).  (5.20)

Obviously, [G : K] < ∞ if and only if [G′ : p(K)] < ∞, and [G′ : H] < ∞ if and only if [G : p^{−1}(H)] < ∞. Thus, the claim follows from (5.19) and (5.20).

5.4 Extensions of free abelian groups

The following is a consequence of Theorem 5.13 (iii) and Corollary 5.18. It has already been proved for crystallographic groups by Connolly-Fehrmann-Hartglass [5].

Example 5.21 (Virtually ℤ^{n}-groups). Let G be a group which contains ℤ^{n} as subgroup of finite index. If n ≥ 2, then

hdim^{G}(\underline{\underline{E}}G) = n + 1.

If n ≤ 1, then hdim^{G}(\underline{\underline{E}}G) = 0.

Theorem 5.22. Let 1 → ℤ^{n} → G ⟶^{p} ℤ → 1 be a group extension for n ≥ 2. Let ϕ: ℤ^{n} → ℤ^{n} be the automorphism given by conjugation with some g ∈ G which is mapped to a generator of ℤ under p.

Then there is the following dichotomy:

(i) We have cd(ker(ϕ^{k} − id)) ≤ 1 for all k ∈ ℤ. In this case,

hdim^{G}(\underline{\underline{E}}G) = n + 1;

(ii) We have cd(ker(ϕ^{k} − id)) ≥ 2 for some k ∈ ℤ. In this case,

hdim^{G}(\underline{\underline{E}}G) = n + 2.

Proof. We first deal with the case that ϕ is periodic, i.e., there exists k ≥ 1 such that ϕ^{k} = id. Since n ≥ 2, we are then in the situation of case (ii). The group G contains a subgroup of finite index which is isomorphic to ℤ^{n+1} and, hence, hdim^{G}(\underline{\underline{E}}G) = n + 2 by Example 5.21. Thus, we can assume in the remainder that ϕ is not periodic.

Next, we show that the epimorphism p: G → ℤ satisfies p(V) = 0 for all [V] · G ∈ [VCY \ FIN]_{f}/G. Suppose to the contrary that there exists [V] ∈ [VCY \ FIN] such that N_{G}[V] has finite index in G and p(V) ≠ 0. Because of Lemma 5.15, we can assume without loss of generality that V is infinite cyclic and N_{G}[V] = N_{G}V. Since C_{G}V has finite index in N_{G}V, we conclude that C_{G}V has finite index in G. Let v ∈ V be a generator of V. Then ϕ^{p(v)} : ℤ^{n} → ℤ^{n} is given by conjugation with v and, hence, is the identity on the subgroup ℤ^{n}∩C_{G}V of finite index in ℤ^{n}. This shows that ϕ^{p(v)} = id, contradicting the assumption that ϕ is not periodic.

Now, consider [V] · G ∈ [VCY \ FIN]_{f}/G. Then p(V) = 0 implies ℤ^{n} ⊆ N_{G}[V] and N_{G}[V] = p^{−1}(p(N_{G}[V])). One easily checks that

p(N_{G}[V]) = {k ∈ ℤ | V ⊆ ker(ϕ^{k} − id) or V ⊆ ker(ϕ^{k} + id)}.

(i) Obviously, ker(ϕ^{k} −id) = 0 holds for all k ∈ ℤ if and only if [VCY \FIN]_{f}/G is empty. In this case, hdim^{G}(\underline{\underline{E}}G) = n + 1 because of Theorem 5.13 (i). We are left to treat the case that there exists n ≥ 1 such that cd(ker(ϕ^{n} − id)) = 1, while cd(ker(ϕ^{k} − id)) ≤ 1 holds for all k ≥ 1. Put V := ker(ϕ^{n} − id). Then the the index of N_{G}V in G is finite, and so [V] · G belongs to [VCY \ FIN]_{f}/G. Consider another element [W] · G ∈ [VCY \ FIN]_{f}/G. There exists k ≥ 1 such that W ⊆ ker(ϕ^{k} − id) or W ⊆ ker(ϕ^{k} + id), hence W ⊆ ker(ϕ^{2kn} − id). Since also V ⊆ ker(ϕ^{2kn}−id) and cd(ker(ϕ^{2kn}−id)) ≤ 1, we conclude [V]·G = [W]·G, and we have shown that [VCY \ FIN]_{f}/G = {[V] · G} contains precisely one element [V] · G.

Now hdim^{G}(\underline{\underline{E}}G) = n + 1 follows from Theorem 5.13 (ii)(b). Namely, any infinite cyclic subgroup C ⊆ ℤ^{n} such that C ∩ V = {1} yields an element [C] · G ∈ [VCY \ FIN]_{f}/G such that cd(N_{G}[C]) = n = cd(G) − 1. (ii) Choose k ≥ 1 with cd(ker(ϕ^{k} −id)) ≥ 2 and put G′ := p^{−1}(kℤ). Then G′ is a subgroup of G of finite index whose center contains ℤ^{2}, and hdim^{G}(\underline{\underline{E}}G) = n+2 follows from Theorem 5.13 (iii).

Theorem 5.23. Let 1 → ℤ^{m} → G ⟶^{p} ℤ^{n} → 1 be a group extension such that cent(G) = ℤ^{m}. Then

hdim^{G}(\underline{E}G) = cd(G) = m + n

and

hdim^{G}(\underline{\underline{E}}G) = \begin{cases}0 & if m + n ≤ 1; \\ n + 1 & if m = 1 and n ≥ 1; \\ m + n + 1 & if m ≥ 2 and n ≥ 1. \end{cases}

Proof. The group G is poly-ℤ by Lemma 5.14 (i). We get hdim^{G}(\underline{E}G) = cd(G) = m + n from Theorem 5.13 and Lemma 5.14 (iv).

If m + n ≤ 1, then G is infinite cyclic, so hdim^{G}(\underline{\underline{E}}G) = 0. If m = 0, then G = ℤ^{n} and the claim follows from Example 5.21. If m ≥ 2, the claim follows from Theorem 5.13 (iii). It remains to treat the case m = 1 and n ≥ 1, in which we want to show for an infinite cyclic subgroup C ⊆ G with [C] ≠ [cent(G)] that cd(N_{G}[C]) = n.

Consider the canonical projection p: G → G/ cent(G) ≅ ℤ^{n}. According to Lemma 5.15, we can assume that N_{G}C = N_{G}[C]. Since C_{G}C has finite index in N_{G}C, it suffices to prove that cd(C_{G}C) = n. Fix a generator z ∈ C and an element g ∈ G. Let c(g): G → G be the automorphism sending g′ to gg′g^{−1}, then p∘c(g)(z) = p(z). There is, therefore, precisely one element σ(g) ∈ cent(G) satisfying c(g)(z) = zσ(g). A straightforward computation for g_{1}, g_{2} ∈ G shows that zσ(g_{1}g_{2}) = zσ(g_{1})σ(g_{2}), which implies σ(g_{1}g_{2}) = σ(g_{1})σ(g_{2}). Moreover, σ(1) = 1. Thus, we have defined a group homomorphism σ: G → cent(G) whose kernel is C_{G}C. As [C] ≠ [cent(G)] by assumption, cent(G) cannot contain C, so ker(σ) ≠ G. This means that σ is non-trivial and that, hence, its image is an infinite cyclic group. We conclude from Lemma 5.14 (iv) that cd(C_{G}C) = n, as we wanted to show.

The above implies that we are in the situation of Theorem 5.13 (ii)(b), and we get hdim^{G}(\underline{\underline{E}}G) = cd(G) = n + 1.

Theorem 5.24. Let 1 → ℤ → H ⟶^{p} ℤ^{n} → 1 be a group extension such that ℤ = cent(H) and n ≥ 2. Consider a group automorphism f: H → H. It sends cent(H) to cent(H). Let \overline{f}: ℤ^{n} → ℤ^{n} be the group automorphism induced by f, i.e., which satisfies p ∘ f = \overline{f} ∘ p. Then G := H ⋊_{f} ℤ satisfies cd(G) = n + 2, we have

hdim^{G}(\underline{E}G) = n + 2,

and precisely one of the following cases occurs:

(i) We have ker(\overline{f}^{k} − id) = 0 for every k ∈ ℤ, k ≠ 0. In this case,

hdim^{G}(\underline{\underline{E}}G) = n + 1;

(ii) We have ker(\overline{f}^{k} − id) ≠ 0 for some k ∈ ℤ, k ≠ 0 and \overline{f} is not periodic, i.e., there is no l ∈ ℤ, l ≥ 1 such that \overline{f}^{l} = id. In this case,

hdim^{G}(\underline{\underline{E}}G) = n + 2;

(iii) The map \overline{f} is periodic. In this case,

hdim^{G}(\underline{\underline{E}}G) = n + 3.

Proof. Let pr: G = H ⋊_{f} ℤ → ℤ be the obvious projection, then G fits into an exact sequence 1 → H → G ⟶^{pr} ℤ → 1. Therefore, hdim^{G}(\underline{E}G) = cd(G) = n+2 follows from Lemma 5.14 (i) and (iv) and Theorem 5.13.

Let V ⊆ G be an infinite cyclic subgroup. Then V ⊆ cent(H) if and only if [V] = [cent(H)] and, in this case, N_{G}[V] = G. Provided that [V] ≠ [cent(H)], we first want to show

cd(N_{G}[V]) = \begin{cases}n & if pr(N_{G}[V]) = 0; \\ n + 1 & if pr(N_{G}[V]) ≠ 0 \\ & and pr(V) = 0; \\ 2 + cd(p(N_{G}[V] ∩ H)) & if pr(V) ≠ 0. \end{cases}  (5.25)

Suppose that pr(V) = 0, i.e., V ⊆ H. We have already seen in the proof of Theorem 5.23 that cd(N_{H}[V]) = n for [V] ≠ [cent(H)]. It follows from the short exact sequence

1 → N_{H}[V] = N_{G}[V] ∩ H → N_{G}[V] → pr(N_{G}[V]) → 1  (5.26)

and Lemma 5.14 (iv) that cd(N_{G}[V]) = n + cd(pr(N_{G}[V])). If pr(N_{G}[V]) = 0, then cd(pr(N_{G}[V])) = 0. If pr(N_{G}[V]) ≠ 0, then pr(N_{G}[V]) ≅ ℤ, which implies cd(pr(N_{G}[V])) = 1.

Now suppose that pr(V) ≠ 0. Then pr(V) is infinite cyclic, so the same must be true for pr(N_{G}[V]). Let w be a generator of cent(H) and v be a generator of V. Since f induces ± id on cent(H), we have vwv^{−1} = w^{±1}. This implies vw^{2}v^{−1} = w^{2}, thus w^{2} ∈ C_{G}V ⊆ N_{G}[V]. Therefore, N_{G}[V] ∩ cent(H) is an infinite cyclic group, and the short exact sequences

1 → N_{G}[V] ∩ cent(H) → N_{G}[V] ∩ H → p(N_{G}[V] ∩ H) → 1

and (5.26) together with Lemma 5.14 (iv) imply cd(N_{G}[V]) = 2 + cd(p(N_{G}[V] ∩ H)). This finishes the proof of (5.25).

Next, we show in the case pr(V) ≠ 0 that

p(N_{G}[V] ∩ H) ⊆ ker(\overline{f}^{k} − id) for some k ∈ ℤ, k ≠ 0.  (5.27)

Because of Lemma 5.15, we can assume without loss of generality that N_{G}[V] = N_{G}V. Recall that C_{G}V has finite index in N_{G}V. Hence it suffices to show that p(C_{G}V ∩ H) ⊆ ker(\overline{f}^{k} − id) if k ∈ pr(V). Let t ∈ G = H ⋊_{f} ℤ be given by the generator of ℤ. Then conjugation with t induces the automorphism f on H, and ht^{k} ∈ V for some h ∈ H. We get for u ∈ C_{G}V ∩ H

\overline{f}^{k}(p(u)) = p ∘ f^{k}(u) = p(t^{k}ut^{−k}) = p(h)p(t^{k}ut^{−k})p(h)^{−1}

= p(ht^{k}u(ht^{k})^{−1}) = p(u),

which finishes the proof of (5.27).

Finally, we show in the case pr(N_{G}[V]) ≠ 0 and pr(V) = 0 that

p(V) ⊆ ker(\overline{f}^{k} − id) for some k ∈ ℤ, k ≠ 0.  (5.28)

Again, we can assume that N_{G}[V] = N_{G}V. Since C_{G}V has finite index in N_{G}V, we have pr(C_{G}[V]) ≠ 0, so we can choose a non-trivial element k ∈ pr(C_{G}[V]). Then ht^{k} ∈ C_{G}V for some h ∈ H, and the same calculation as above shows that \overline{f}^{k}(p(v)) = p(v) holds for v ∈ V. This finishes the proof of (5.28). Now we are ready to prove the assertions appearing in Theorem 5.24.

(i) Consider an infinite cyclic subgroup V ⊆ G such that [V] ≠ [cent(H)]. We first deal with the case pr(V) = 0. Then pr(N_{G}[V]) = 0 since, otherwise, (5.28) would yield V ⊆ cent(H). Using (5.25), the conclusion is that cd(N_{G}[V]) = n. Since cd(G) = n + 2, we get hdim^{G}(\underline{\underline{E}}G) = n + 1 from Theorem 5.13 (ii)(a). It remains to consider the case pr(V) ≠ 0. Then we conclude cd(N_{G}[V]) = 2 + cd(p(N_{G}V ∩ H)) from (5.25). Because of (5.27) we get cd(N_{G}[V]) = 2. Since n ≥ 2, we have vcd(G) ≥ 4. Now hdim^{G}(\underline{\underline{E}}G) = n + 1 follows from Theorem 5.13 (ii)(a).

(ii) We claim that there exists an infinite cyclic subgroup V ⊆ H such that [V] ≠ [cent(H)], pr(V) = 0 and pr(N_{G}V) ≠ 0. To do so, we choose an even k ∈ ℤ such that ker(\overline{f}^{k} − id) ≠ 0, an element h_{0} ∈ H such that 0 ≠ p(h_{0}) ∈ ker(\overline{f}^{k} − id), and define V to be the infinite cyclic group generated by h_{0}. Then p(f^{k}(h_{0})) = \overline{f}^{k} ∘ p(h_{0}) = p(h_{0}). Hence, we can find z_{0} ∈ cent(H) = ℤ such that t^{k}h_{0}t^{−k} = f^{k}(h_{0}) = h_{0}z_{0}. Since k is even, t^{k}z_{0}t^{−k} = z_{0}. As h_{0} ∉ cent(H), we can find h_{1} ∈ H such that h_{1}h_{0}h_{1}^{−1} ≠ h_{0}. However, p(h_{1}h_{0}h_{1}^{−1}) = p(h_{1})p(h_{0})p(h_{1}^{−1}) = p(h_{0}) and so h_{1}h_{0}h_{1}^{−1} = h_{0}z_{1} for some non-trivial z_{1} ∈ cent(H). Choose integers m_{0} ≠ 0 and m_{1} such that z_{0}^{m_{0}} = z_{1}^{m_{1}}, and put u = h_{1}^{−m_{1}}t^{m_{0}k}. From t^{m_{0}k}h_{0}t^{−m_{0}k} = h_{0}z_{0}^{m_{0}} and h_{1}^{−m_{1}}h_{0}h_{1}^{m_{1}} = h_{0}z_{1}^{−m_{1}} we then conclude uh_{0}u^{−1} = h_{0} and pr(u) = km_{0}. This finishes the proof of the above claim. It follows from (5.25) that cd(N_{G}[V]) = n + 1.

Since cd(ker(\overline{f}^{k} − id)) = n implies \overline{f}^{k} = id but \overline{f} is not periodic, we have cd(ker(\overline{f}^{k}−id)) ≤ n−1 for every k ∈ ℤ with k ≠ 0. Thus, (5.25) and (5.27) show that cd(N_{G}[V]) ≤ n + 1 holds for all infinite cyclic subgroups V ⊆ G satisfying [V] ≠ [cent(H)]. Theorem 5.13 (ii)(b) now implies hdim^{G}(\underline{\underline{E}}G) = n + 2.

(iii) We want to show the existence of a finite index subgroup of G whose center contains ℤ^{2}. Then hdim^{G}(\underline{\underline{E}}G) = n + 3 follows from Theorem 5.13 (iii). Let us begin by choosing an even k ∈ ℤ such that \overline{f}^{k} = id. We define a group homomorphism σ_{f^{k}}: H → cent(H) = ℤ by σ_{f^{k}}(h) := h^{−1}f^{k}(h) for h ∈ H. Since k is even, the restriction of f^{k} to cent(H) = ℤ is the identity, which implies that the restriction of σ_{f^{k}} to cent(H) is trivial. Hence σ_{f^{k}} factorizes through the projection p: H → H/ cent(H) = ℤ^{n}, yielding homomorphism \overline{σ}_{f^{k}}: ℤ^{n} → ℤ. We have f^{k}(h) = h\overline{σ}_{f^{k}}(p(h)) for h ∈ H.

Conjugation with h_{0} ∈ H induces a group automorphism c(h_{0}): H → H whose restriction to cent(H) is the identity and for which the induced automorphism \overline{c(h_{0})} of H/ cent(H) = ℤ^{n} is the identity. Just as above for f^{k}, we define a homomorphism \overline{σ}_{c(h_{0})} : ℤ^{n} → ℤ such that c(h_{0})(h) = h_{0}hh_{0}^{−1} = h\overline{σ}_{c(h_{0})}(p(h)) holds for h ∈ H, and one easily checks that

\overline{σ}_{c(h_{0}h_{1})} = \overline{σ}_{c(h_{0})∘c(h_{1})} = \overline{σ}_{c(h_{0})} + \overline{σ}_{c(h_{0})}

for h_{0}, h_{1} ∈ H. That way, we obtain an injective group homomorphism

\overline{σ}: H/ cent(H) → hom_{ℤ}(H/ cent(H), ℤ), \overline{h} ↦ \overline{σ}_{c(h)}.

Since its source and target are isomorphic to ℤ^{n}, its image has finite index. As \overline{σ}_{f^{lk}} = l · \overline{σ}_{f^{k}} holds for all l ∈ ℤ, it follows that we can find an integer l ≥ 1 and h ∈ H such that \overline{σ}_{f^{lk}} = \overline{σ}_{c(h)}, which implies f^{kl} = c(h).

The subgroup pr^{−1}(kl ·ℤ) ⊆ G has finite index in G and is isomorphic to the semidirect product H ⋊_{f^{kl}} ℤ with respect to the automorphism f^{kl}: H → H. Since f^{kl} is an inner automorphism, pr^{−1}(kl · ℤ) is isomorphic to the direct product H × ℤ and, hence, contains cent(H) × ℤ ≅ ℤ^{2} in its center. This finishes the proof of Theorem 5.24.

Example 5.29 (Three-dimensional Heisenberg group). The three-dimen-sional Heisenberg group is the group given by the presentation Hei = ⟨u, v, z | [u, v] = z, [u, z] = 1, [v, z] = 1⟩. Define automorphisms f_{k}: Hei → Hei for k = −1, 0, 1 by

f_{−1}(u) = u^{3}v,  f_{−1}(v) = u^{2}v,  f_{−1}(z) = z;
f_{0}(u) = u,  f_{0}(v) = uv,  f_{0}(z) = z;
f_{1}(u) = u,  f_{1}(v) = v,  f_{1}(z) = z.

Put G = Hei ⋊_{f_{k}}ℤ for k = −1, 0, 1. Then Theorem 5.24 implies

hdim^{G_{k}}(\underline{E}G_{k}) = 4;
hdim^{G_{k}}(\underline{\underline{E}}G_{k}) = 4 + k.

5.5 Models for E_{SFG}(G)

Let SFG be the family of subgroups of G which are contained in some finitely generated subgroup, i.e., H ∈ SFG if and only if there exists a finitely generated subgroup H′ ⊆ G such that H ⊆ H′. This is the smallest family of subgroups of G which contains all finitely generated subgroups of G.

Example 5.30. Suppose that G is countable. Then there exists a one-di-mensional model for E_{SFG}(G) because of Theorem 4.3 and Lemma 4.2. In particular, there exists a one-dimensional model for \underline{E}G or \underline{\underline{E}}G if G is locally finite or locally virtually cyclic, respectively.

Theorem 5.31. Let G be an infinite group of cardinality ℵ_{n}. Let SFG be the family of subgroups defined above. Then

hdim^{G}(E_{SFG}(G)) ≤ n + 1.

Proof. We use induction on n ∈ ℕ. The case n = 0 has already been settled in Example 5.30. The induction step from n − 1 to n ≥ 1 is done as follows.

We can write G = ⋃_{α<ω_{n}} G_{α} for subgroups G_{α} of cardinality ℵ_{n−1} such that G_{α} ⊆ G_{β} if α ≤ β. By induction hypothesis, for α < ω_{n} there are models X_{α} for E_{SFG}(G_{α}) of dimension n. The induction step now must provide us with an (n + 1)-dimensional model for E_{SFG}(G). If we set G_{ω_{n}} := G, this will be accomplished by using transfinite induction for α ≤ ω_{n} to construct (n + 1)- dimensional models Y_{α} for E_{SFG}(G_{α}) such that G_{γ} ×_{G_{α}} Y_{α} ⊆ G_{γ} ×_{G_{β}} Y_{β} is a G_{γ}-subcomplex if α ≤ β ≤ γ.

Let Y_{0} := X_{0}. Now suppose that α has got a predecessor. Then the universal property of E_{SFG}(G_{α}) yields a G_{α}-map f_{α}: G_{α} ×_{G_{α−1}} X_{α−1} → X_{α}, which we can assume to be cellular by the equivariant cellular approximation theorem. We define Y_{α} by the G_{α}-pushout

\xymatrix{G_{α} ×_{G_{α−1}} X_{α−1} \ar[d]^{id ×g_{α−1}} \ar[r]^{i_{α}} & cyl(f_{α}) \ar[d] \\ G_{α} ×_{G_{α−1}} Y_{α−1} \ar[r] & Y_{α}}

in which i_{α} is the obvious inclusion into the mapping cylinder of f_{α} and g_{α−1} the up to G_{α−1}-homotopy unique homotopy equivalence which comes from the universal property of E_{SFG}(G_{α−1}). Hence, Y_{α} is G_{α}-homotopy equivalent to X_{α} and therefore a model for E_{SFG}(G_{α}). Moreover, Y_{α} is clearly (n + 1)- dimensional. Finally, if α is a limit ordinal, we define Y_{α} to be the union of the G_{α} ×_{G_{β}} Y_{β} for β < α.

Example 5.32. Let G be an infinite locally finite group of cardinality ℵ_{n}. Then it is consistent with ZFC (Zermelo-Fraenkel plus axiom of choice) that

hdim^{G}(\underline{E}G) = n + 1.

The existence of an (n+1)-dimensional model for \underline{E}G follows from Theorem 5.31 and has already been proved by Dicks-Kropholler-Leary [8, Theorem 2.6]. Now the claim follows from the fact that it is consistent with ZFC that the cohomological dimension over ℚ of G satisfies cd_{ℚ}(G) = n + 1 (see [12, Theorem A]).

We emphasize that the assumption in Example 5.30 that G is countable is necessary. Namely, Dunwoody [9, Theorem 1.1] has proved that for a group K there is a one-dimensional model for \underline{E}K if and only if cd_{ℚ}(K) ≤ 1.

5.6 Low dimensions

For countable groups G, the question whether there are models for \underline{E}G and \underline{\underline{E}}G of small dimension has the following answer. The assumption that G is countable is essential (see Example 5.32).

Theorem 5.33. (i) Let G be a countable group which is locally virtually cyclic. Then

hdim^{G}(\underline{E}G) = \begin{cases}0 & if G is finite; \\ 1 & if G is infinite and either locally finite \\ & or virtually cyclic; \\ 2 & otherwise, \end{cases}

and

hdim^{G}(\underline{\underline{E}}G) = \begin{cases}0 & if G is virtually cyclic; \\ 1 & otherwise; \end{cases}

(ii) Let G be a countable group satisfying hdim^{G}(\underline{E}G) ≤ 1. Then

hdim^{G}(\underline{\underline{E}}G) = \begin{cases}0 & if G is virtually cyclic; \\ 1 & if G is locally virtually cyclic but \\ & not virtually cyclic; \\ 2 & otherwise. \end{cases}

Proof. (i) The claim about hdim^{G}(\underline{\underline{E}}G) follows from Example 5.30.

We conclude hdim^{G}(\underline{E}G) ≤ 2 from hdim^{G}(\underline{\underline{E}}G) ≤ 1 and Corollary 5.4 (i). If G is locally finite, then hdim^{G}(\underline{E}G) ≤ 1 has already been proved in Example 5.30. Let G be a countable locally virtually cyclic group and hdim^{G}(\underline{E}G) ≤ 1. It remains to show that G is virtually cyclic or locally finite. Assume that G is not locally finite, then we can choose choose a sequence of infinite virtually cyclic subgroups V_{0} ⊆ V_{1} ⊆ V_{2} ⊆ … such that G = ⋃_{i≥0} V_{i}, and we must show that G itself is virtually cyclic.

We begin with the case that each V_{i} is an infinite virtually cyclic group of type I. Type I means that we can find an epimorphism p_{i}: V_{i} → C_{i} onto an infinite cyclic group C_{i}. Let F_{i} be the kernel of p_{i}, which is a finite group. Thus we obtain a nested sequence of inclusions of finite groups F_{1} ⊆ F_{2} ⊆ … and a sequence of inclusions of infinite cyclic groups C_{1} ⊆ C_{2} ⊆ … such that there exists a short exact sequence 1 → F_{i} → V_{i} → C_{i} → 1 for each i.

Now we deal with the hardest step in the proof, where we show that the sequence F_{0} ⊆ F_{1} ⊆ F_{2} ⊆ … is eventually stationary. Let t ∈ V_{0} be an element which is mapped to a generator under the epimorphism p_{0} : V_{0} → C_{0}. If we consider t as an element in V_{i}, conjugation with t induces an automorphism ϕ_{i}: F_{i} → F_{i}. Obviously, the restriction of ϕ_{i} to F_{i−1} is ϕ_{i−1}. We obtain a nested sequence of virtually cyclic groups F_{0} ⋊ ℤ ⊆ F_{1} ⋊ ℤ ⊆ F_{2} ⋊ ℤ ⊆ …, where the i-th semi-direct product F_{i} ⋊ ℤ is to be understood with respect to ϕ_{i}. Since K := ⋃_{i≥0} F_{i} ⋊ ℤ is a subgroup of G, it satisfies hdim^{K}(\underline{E}K) ≤ 1. This implies H_{K}^{2}(\underline{E}K; M) = 0 for any ℚ[K]-module M.

The ℚ[K]-chain map C_{∗}(EK) → C_{∗}(\underline{E}K) of the cellular ℚ[K]-chain complexes coming from a K-map EK → \underline{E}K induces an isomorphism on homology. Since both of these ℚ[K]-chain complexes are projective, this map is a ℚ[K]- chain homotopy equivalence. Hence, we obtain for every ℚ[K]-module M an isomorphism

H^{k}(K; M) = H^{k}(hom_{ℚ[K]}(C_{∗}(EK), M)) ≅ H^{k}(hom_{ℚ[K]}(C_{∗}(\underline{E}K), M))

= H_{K}^{k}(\underline{E}K; M).

In particular, H^{2}(K; M) = 0.

If we put F = ⋃_{i≥0} F_{i}, then the collection of the ϕ_{i}-s yields an automorphism ϕ: F → F, with respect to which K = F ⋊ℤ. Let M be any ℚ[K]-module. The Hochschild-Serre spectral sequence associated to 1 → F → K = F ⋊ℤ → ℤ → 1 yields the exact sequence

H^{1}(F; M) ⟶^{id −ϕ_{∗}} H^{1}(F; M) → H^{2}(K; M) → H^{2}(F; M)

where the automorphism ϕ_{∗} of H^{1}(F; M) is the one induced by ϕ: F → F and the ℚ[F ⋊ ℤ]-structure on M. Hence

H^{2}(K; M) ≅ coker (id −ϕ_{∗}: H^{1}(F; M) → H^{1}(F; M))

since we will show in a moment that H^{2}(F; M) = 0.

Next, we compute H^{∗}(F; M) for a given ℚ[F]-module M. We have already explained the isomorphism

H^{1}(F; M) = H^{1}(hom_{ℚ[F]}(C_{∗}(EF), M)) ≅ H^{1}(hom_{ℚ[F]}(C_{∗}(\underline{E}F), M)).

It follows from Lemma 4.5 that we obtain an F -pushout

\xymatrix{∐_{i≥0} F/F_{i} × S^{0} \ar[d] \ar[r]^{∐_{i≥0} k_{i}} & ∐_{i≥0} F/F_{i} \ar[d] \\ ∐_{i≥0} F/F_{i} × D^{1} \ar[r] & \underline{E}F}

where the left vertical arrow is the canonical inclusion and k_{i}: F/F_{i} × S^{0} → ∐_{i≥0} F/F_{i} sends (u, −1) to u ∈ F/F_{i} and (u, 1) to the image of u under the projection F/F_{i} → F/F_{i+1}. Hence, the cellular ℚ[F]-chain complex of \underline{E}F is concentrated in dimensions 0 and 1 and is given there by

⨁_{i≥0} ℚ[F/F_{i}] \underrightarrow{⨁_{i≥0} l_{i}} ⨁_{i≥0} ℚ[F/F_{i}],

where l_{i} sends u ∈ ℚ[F/F_{i}] to the difference of the element given by u itself and the element given by the image of u under ℚ[pr_{i}]: ℚ[F/F_{i}] → ℚ[F/F_{i+1}] for pr_{i}: F/F_{i} → F/F_{i+1} the canonical projection. Therefore, H^{k}(F; M) = 0 for k ≥ 2, whereas H^{1}(F; M) is the cokernel of the ℚ[F]-homomorphism

δ_{M}: ∏_{i≥0} M^{F_{i}} → ∏_{i≥0} M^{F_{i}}, (x_{i})_{i≥0} ↦ (x_{i} − x_{i+1})_{i≥0}.

Let us show that H^{1}(F; ℚ[F]) = 0 if and only if the sequence F_{0} ⊆ F_{1} ⊆ F_{2} ⊆ … is eventually stationary. If it becomes stationary, then F is finite, so H^{1}(F; ℚ[F]) = 0. Suppose that H^{1}(F; ℚ[F]) = 0 and let N_{F_{i}} ∈ ℚ[F]^{F_{i}} be the element ∑_{f∈F_{i}} f. Since δ_{ℚ[F]} is surjective, there exists (x_{i})_{i≥0} ∈ ∏_{i≥0} ℚ[F]^{F_{i}} satisfying N_{F_{i}} = x_{i} − x_{i+1}. Thus, we get for i ≥ 1

x_{i} = x_{0} − N_{G_{i−1}} − N_{G_{i−2}} − ⋯ − N_{G_{0}}.

After choosing i_{0} in such a way that x_{0} ∈ ℚ[F_{i_{0}}], we get that x_{i} ∈ ℚ[F_{i−1}] holds for for i > i_{0}. Since x_{i} ∈ ℚ[F]^{F_{i}}, we conclude that F_{i} = F_{i−1} holds for i > i_{0}, which means that F_{0} ⊆ F_{1} ⊆ F_{2} ⊆ … is eventually stationary. For more information about H^{∗}(F; ℚ[F]) for locally finite groups F, we refer for instance to [8].

Now suppose that M is a ℚ[K]-module. We obtain the commutative diagram

\xymatrix{∏_{i≥0} M^{F_{i}} \ar[d]_{id −l_{t}} \ar[r]_{δ_{M}} & ∏_{i≥0} M^{F_{i}} \ar[d]_{id −l_{t}} \ar[r] & H^{1}(F; M) \ar[d]_{id −ϕ_{∗}} \ar[r] & 0 \\ ∏_{i≥0} M^{F_{i}} \ar[r]_{δ_{M}} & ∏_{i≥0} M^{F_{i}} \ar[r] & H^{1}(F; M) \ar[r] & 0}

where l_{t}: M → M is multiplication with the generator of ℤ considered as an element in K = F ⋊ ℤ.

Specializing to M = ℚ[K] = ℚ[F ⋊ ℤ], we can write down the following exact sequence

0 → ℚ[F ⋊ ℤ] ⟶^{id −l_{t}} ℚ[F ⋊ ℤ] ⟶^{ϵ} ℚ[F] → 0,

where ϵ sends ∑_{n∈ℤ} r_{i} · f_{i}t^{i} to ∑_{n∈ℤ} r_{i} · ϕ^{−i}(f_{i}) for r_{i} ∈ ℚ, f_{i} ∈ F and t ∈ ℤ a fixed generator. The composition

∏_{i≥0} ℚ[F ⋊ ℤ]^{F_{i}} ⟶^{δ_{ℚ[F ⋊ℤ]}} ∏_{i≥0} ℚ[F ⋊ ℤ]^{F_{i}} ⟶^{ϵ} ∏_{i≥0} ℚ[F]^{F_{i}}

agrees with the composition

∏_{i≥0} ℚ[F ⋊ ℤ]^{F_{i}} ⟶^{ϵ} ∏_{i≥0} ℚ[F]^{F_{i}} ⟶^{δ_{ℚ[F]}} ∏_{i≥0} ℚ[F]^{F_{i}}

since ϵ ∘ l_{t} = ϵ. Hence we obtain an isomorphism

H^{1}(F; ℚ[F]) ≅ coker(δ_{ℚ[F]} : H^{1}(F; ℚ[F]) → H^{1}(F; ℚ[F]))

≅ coker(id −ϕ_{∗}: H^{1}(F; ℚ[F ⋊ ℤ]) → H^{1}(F; ℚ[F ⋊ ℤ])) ≅ H^{2}(K; ℚ[K]).

As H^{2}(K; ℚ[K]) is trivial, the same holds for H^{1}(F; ℚ[F]). We have already shown that this implies that F = ⋃_{i≥0} F_{i} is finite.

Returning to the beginning of the proof and denoting by A the colimit of the system C_{0} ⊆ C_{1} ⊆ C_{2} ⊆ … of infinite cyclic groups, we obtain the exact sequence 1 → F → G → A → 1. Since hdim^{G}(\underline{E}G) ≤ 1 by assumption, the cohomological dimension of G over ℚ satisfies cd_{ℚ}(G) ≤ 1. Consequently, also cd_{ℚ}(A) ≤ 1 as F is finite. By a result of Dunwoody [9, Theorem 1.1] already mentioned in Example 5.32, this means that hdim^{A}(\underline{E}A) ≤ 1. However, as A is torsionfree, this implies that A is free (see [18] and [19]). Since ℤ is the only non-trivial abelian free group, A is infinite cyclic, so G is virtually cyclic. This finishes the proof in the case that each V_{i} is infinite cyclic of type I.

We are left to treat the case that there exists i_{0} such that V_{i_{0}} is infinite virtually cyclic of type II, i.e., there exists an epimorphism V_{i_{0}} → D_{∞} onto the infinite dihedral group. Since V_{i_{0}} is a subgroup of V_{i} for i ≥ i_{0}, the virtually cyclic group V_{i} is of type II for i ≥ i_{0} or, equivalently, H_{1}(V_{i}) is finite for i ≥ i_{0}. Let W_{i} be the commutator of V_{i}. Then we obtain a sequence of virtually cyclic subgroups W_{0} ⊆ W_{1} ⊆ W_{2} ⊆ … such that W_{i} is of type I and has finite index in V_{i} for i ≥ i_{0}. Let W be the subgroup ⋃_{i≥0} W_{i} of G. Since dim^{G}(\underline{E}G) ≤ 1 by assumption, we have hdim^{W} (\underline{E}W) ≤ 1 as well. Thus, by what we have already proved above, W is virtually cyclic and, in particular, finitely generated. Hence, the sequence W_{0} ⊆ W_{1} ⊆ W_{2} ⊆ … is eventually stationary, and we can assume without loss of generality that W = W_{i} for all i ≥ 0. This implies that each map H_{1}(V_{i}) → H_{1}(V_{i+1}) is injective. Putting A := ⋃_{i≥0} H_{1}(V_{i}), we obtain an exact sequence 1 → W → G → A → 0, where W is an infinite virtually cyclic group of type I. We are going to show that A is finite.

Let K denote the kernel of the canonical epimorphism W → C for C = H_{1}(W)/ tors(H_{1}(W)). Then K is a finite characteristic subgroup of W and hence normal in G, so there is an exact sequence 1 → K → G → G/K → 1. Note that we have already explained above (using [9, Theorem 1.1]) why hdim^{G}(\underline{E}G) ≤ 1 implies hdim^{G/K}(\underline{E}(G/K)) ≤ 1. Furthermore, there is an exact sequence 1 → C → G/K → A → 1. Let A → aut(C) be the associated conjugation homomorphism, let A′ ⊆ A be its kernel, and let G′ the preimage of A′ under the epimorphism G/K → A. Then we obtain a central extension 1 → C → G′ → A′ → 1 such that hdim^{G′}(\underline{E}G′) ≤ 1 and A′ ⊆ A is a subgroup of finite index. We have A′ = ⋃_{i≥0}(A′ ∩ H_{1}(V_{i})). The computations appearing in the proof above imply that either A′ is a finite abelian group or H^{1}(A′; ℚ[A′]) ≅ coker(δ_{ℚ[A′]}) ≠ 0, and that H^{k}(A′; ℚ[A′]) = 0 for k ≥ 2. From the Hochschild-Serre cohomology spectral sequence applied to 1 → C → G′ → A′ → 1 we obtain an isomorphism H^{1}(A′; ℚ[A′]) ≅ H^{2}(G′; ℚ[A′]), and the latter term vanishes as hdim^{G/K}(\underline{E}(G/K)) ≤ 1. We conclude that A′ and hence A are finite, showing that G is virtually cyclic. This finishes the proof of assertion (i). (ii) First we show that hdim^{G}(\underline{\underline{E}}G) ≤ 2 provided that hdim^{G}(\underline{E}G) ≤ 1 and G is countable. Consider an infinite virtually cyclic subgroup V ⊆ G. Let H ⊆ N_{G}[V] be a finitely generated subgroup such that V ⊆ H. From hdim^{G}(\underline{E}G) ≤ 1 we get hdim^{H}(\underline{E}H) ≤ 1, and since H is finitely generated, this implies that it is virtually finitely generated free (see [11, Theorem 1]). In particular, H is word-hyperbolic, so we conclude from Theorem 3.1 and Example 3.6 that N_{H}[V] is virtually cyclic. Since H = H ∩ N_{G}[V] = N_{H}[V], this means that H is virtually cyclic. Thus, we have shown that N_{G}[V] is locally virtually cyclic, and as hdim^{G}(\underline{E}G) ≤ 1 implies hdim^{N_{G}[V]}(\underline{E}N_{G}[V]) ≤ 1, assertion (i) shows that N_{G}[V] is locally finite or virtually cyclic. In fact, N_{G}[V] contains the infinite virtually cyclic group V, so N_{G}[V] is virtually cyclic. This implies that VCY[V] consists of all subgroups of N_{G}[V] and, hence, {•} is a model for E_{VCY[V]}(N_{G}[V]). Summarizing, we can arrange that dim(\underline{E}N_{G}[V]) ≤ 1 and dim(E_{VCY[V]}(N_{G}[V])) ≤ 1 for every infinite virtually cyclic subgroup V of G. Now Theorem 2.3 together with Remark 2.5 implies hdim^{G}(\underline{\underline{E}}G) ≤ 2.

If G is countable and locally virtually cyclic, then hdim^{G}(\underline{\underline{E}}G) ≤ 1 has already been proved in assertion (i). It remains to show that G is locally virtually cyclic provided that hdim^{G}(\underline{E}G) and hdim^{G}(\underline{\underline{E}}G) are less or equal to 1. Let G_{0} ⊆ G be a finitely generated subgroup of G. Then hdim^{G}(\underline{E}G_{0}) and hdim^{G}(\underline{\underline{E}}G_{0}) are less or equal to 1. This implies that G_{0} contains a finitely generated free subgroup G_{1} of finite index (see [11, Theorem 1]). The group G_{1} satisfies NM_{FIN ⊆VCY} by Example 3.6, so from Corollary 2.11 we get a cellular G_{1}-pushout

\xymatrix{∐_{V ∈M} G_{1} ×_{V} \underline{E}V \ar[d]^{∐_{V ∈M} id_{G_{1}} ×_{V} i_{V}} \ar[r]^{i} & \underline{E}G_{1} \ar[d] \\ ∐_{V ∈M} G_{1} ×_{V} cone(\underline{E}V) \ar[r] & \underline{\underline{E}}G_{1}}

where M is a complete system of representatives of the conjugacy classes of maximal infinite virtually cyclic subgroups of G_{1} and i_{V} is the obvious inclusion. Dividing out the G_{1}-action and then taking the associated Mayer-Vietoris exact sequence yields an injection

⨁_{V ∈M} H_{1}(V \\underline{E}V) → H_{1}(G_{1}\\underline{E}G_{1})

because hdim^{G_{1}}(\underline{\underline{E}}G_{1}) ≤ 1. Since G_{1} is finitely generated free, H_{1}(G_{1}\\underline{E}G_{1}) is finitely generated free and H_{1}(V \\underline{E}V) is isomorphic to ℤ for every V ∈ M. This implies that M is finite, i.e., G_{1} contains only finitely many infinite cyclic subgroups up to conjugacy. Thus, G_{1} is either trivial or infinite cyclic, showing that G_{0} is virtually cyclic. Hence G is locally virtually cyclic. This finishes the proof of Theorem 5.33.

Example 5.34. For the prime number p, let ℤ[1/p] be the subgroup of ℚ consisting of rational numbers x ∈ ℚ for which p^{n} · x ∈ ℤ for some positive integer n (see Example 2.6). Every finitely generated subgroup of ℤ[1/p] is trivial or infinite cyclic. Thus, Theorem 5.33 implies

hdim^{ℤ[1/p]}(\underline{E}ℤ[1/p]) = 2;
hdim^{ℤ[1/p]}(\underline{\underline{E}}ℤ[1/p]) = 1.

Remark 5.35. Let G be a countable group. Then Theorem 5.33 implies that G is infinite locally finite if and only if hdim^{G}(\underline{E}G) = hdim^{G}(\underline{\underline{E}}G) = 1 holds.

6 Equivariant homology of relative assembly maps

Let H_{∗}^{?} be an equivariant homology theory in the sense of [14, Section 1]. Our main example is the equivariant homology theory H_{∗}^{?}(−; K_{R}) appearing in the K-theoretic Farrell-Jones conjecture, where R is a ring (here, rings are always assumed to be associative and unital). It has the property that

H_{n}^{G}(G/H; K_{R}) = H_{n}^{H}({•}; K_{R}) = K_{n}(RH)

holds for every subgroup H ⊆ G, where K_{n}(RH) denotes the algebraic Ktheory of the group ring RH. The Farrell-Jones conjecture for a group G and a ring R says that the assembly map, which is the map induced by the projection \underline{\underline{E}}G → G/G, is an isomorphism

H_{n}^{G}(\underline{\underline{E}}G; K_{R}) ⟶^{≅} H_{n}^{G}(G/G; K_{R}) = K_{n}(RG)

for all n ∈ ℤ.

Bartels [2] has shown that for every group G, every ring R, and every n ∈ ℤ the relative assembly map

H_{n}^{G}(\underline{E}G; K_{R}) → H_{n}^{G}(\underline{\underline{E}}G; K_{R})

is split-injective. Hence, the source of the assembly map appearing in the Farrell-Jones conjecture can be computed in two steps, involving the computation of H_{n}^{G}(\underline{E}G; K_{R}) and the computation of the remaining term which we denote by H_{n}^{G}(\underline{E}G → \underline{\underline{E}}G; K_{R}). Rationally, H_{n}^{G}(\underline{E}G; K_{R}) can be computed using equivariant Chern characters (see [14, Section 1]) or with the help of nice models for \underline{E}G.

The goal of this section is to give some information about H_{n}^{G}(\underline{E}G → \underline{\underline{E}}G; K_{R}). Namely, using the induction structure on H_{∗}^{?}, Corollary 2.10 implies:

Corollary 6.1. Let H_{∗}^{?} be an equivariant homology theory. Suppose that the group G satisfies (M_{FIN ⊆VCY}). Let M be a complete system of representatives of the conjugacy classes of maximal infinite virtually cyclic subgroups of G. Then, for n ∈ ℤ, we obtain an isomorphism

⨁_{V ∈M} H_{n}^{N_{G}V} (\underline{E}N_{G}V → EW_{G}V) ⟶^{≅} H_{n}^{G}(\underline{E}G → \underline{\underline{E}}G).

Remark 6.2. The term H_{n}^{N_{G}V} (\underline{E}N_{G}V → EW_{G}V) appearing in Corollary 6.1 can be analyzed further.

Namely, by assigning to a free W_{G}V -CW -complex X the ℤ-graded abelian group H_{∗}^{N_{G}V} (\underline{E}N_{G}V × X → X), we obtain a W_{G}V -homology theory for free W_{G}V -CW -complexes. Here, we consider X as an N_{G}V -CW -complex by restriction with the projection N_{G}V → W_{G}V, equip \underline{E}N_{G}V × X with the diagonal N_{G}V -action and let \underline{E}N_{G}V × X → X be the projection onto X. There is an Atiyah-Hirzebruch spectral sequence converging to H_{p+q}^{N_{G}V} (\underline{E}N_{G}V × X → X) whose E^{2}-term is

E_{p,q}^{2} = H_{p}^{W_{G}V} (EW_{G}V; H_{q}^{N_{G}V} (EN_{G}V × W_{G}V → W_{G}V)),

where the right ℤ[W_{G}V]-module structure on H_{q}^{N_{G}V} (\underline{E}N_{G}V × W_{G}V → W_{G}V) comes from the obvious right W_{G}V -action on W_{G}V and the trivial W_{G}V -action on \underline{E}N_{G}V.

For any N_{G}V -space Y, the map N_{G}V ×_{V} res_{N_{G}V}^{V} Y → Y × W_{G}V given by sending (g, y) to (gy, gV) is an N_{G}V -homeomorphism. In particular, since res_{N_{G}V}^{V} \underline{E}N_{G}V is a model for \underline{E}V, we obtain an N_{G}V -homeomorphism N_{G}V ×_{V} \underline{E}V → \underline{E}N_{G}V × W_{G}V. Together with the induction structure, this induces an isomorphism of abelian groups

H_{q}^{N_{G}V} (EN_{G}V × W_{G}V → W_{G}V)) ≅ H_{q}^{V} (\underline{E}V → \{•\}),

which becomes an isomorphism of ℤ[N_{G}V]-modules if we equip the target with the following W_{G}V -action.

Given \overline{g} ∈ W_{G}V, choose a preimage g ∈ N_{G}V under the projection N_{G}V → W_{G}V. Conjugation with g yields a group homomorphism c(g): V → V. The induction structure yields an isomorphism H_{q}^{V} (\underline{E}V → \{•\}) → H_{q}^{V} (ind_{c(g)} \underline{E}V → ind_{c(g)}\{•\}). Let f: ind_{c(g)} \underline{E}V → \underline{E}V be the V -homotopy equivalence which is unique up to V -homotopy. Obviously, ind_{c(g)}{•} = {•}. Hence, f induces a homomorphism f_{q}: H_{q}^{V} (ind_{c(g)} \underline{E}V → ind_{c(g)}\{•\}) → H_{q}^{V} (\underline{E}V → \{•\}). The composition of these two homomorphisms is an automorphism of H_{q}^{V} (\underline{E}V → \{•\}), which we define to be multiplication with \overline{g}. It is easy to check that this definition is independent of the choice of the preimage g of \overline{g}, and that it defines a W_{G}V -action on H_{q}^{V} (\underline{E}V → \{•\}) which is compatible with the one on H_{q}^{N_{G}V} (EN_{G}V × W_{G}V → W_{G}V).

Example 6.3. Let us consider the special case that the equivariant homology theory is H_{∗}^{?}(−; K_{R}), the one appearing in the K-theoretic Farrell-Jones conjecture. Let V be an infinite virtually cyclic group. If V is of type I, we can write V as a semi-direct product F ⋊ ℤ, and H_{∗}^{V} (\underline{E}V → \{•\}; K_{R}) is the non-connective version of Waldhausen’s Nil-term associated to this semi-direct product (see [1, Section 9]). If V is of type II, then it can be written as an amalgamated product V_{1} ∗_{V_{0}} V_{2} of finite groups, where V_{0} has index two in both V_{1} and V_{2}. In this case, H_{∗}^{V} (\underline{E}V → \{•\}; K_{R}) is the non-connective version of Waldhausen’s Nil-term associated to this amalgamated product (see [1, Section 9]). The W_{G}V -action on these Nil-terms comes from the action of N_{G}V on V by conjugation and the fact that inner automorphisms of groups induce the identity on algebraic K-groups associated to group rings.

In the case of L-theory, the terms H_{∗}^{V} (\underline{E}V → \{•\}; L_{R}^{⟨−∞⟩}) vanish if V is of type I and are given by UNil-terms if V is of type II (see [15, Lemma 4.2]).

Example 6.4. Let H_{∗}^{?} be an equivariant homology theory. Suppose that the group G satisfies (NM_{FIN ⊆VCY}). Let M be a complete system of representatives of the conjugacy classes of maximal infinite virtually cyclic subgroups of G. Then, for n ∈ ℤ, Corollary 6.1 yields an isomorphism

⨁_{V ∈M} H_{n}^{V} (\underline{E}V → \{•\}) ⟶^{≅} H_{n}^{G}(\underline{E}G → \underline{\underline{E}}G).

Now assume in addition that G is torsionfree. If the equivariant homology theory is H_{∗}^{?}(−, K_{R}), then H_{∗}^{V} (\underline{E}V → \{•\}; K_{R}) reduces to NK_{n}(R) ⊕ NK_{n}(R). Here, NK_{n}(R) is the n-th Bass-Nil-group which is defined as the cokernel of the obvious split injection K_{n}(R) → K_{n}(R[t]). In the case of L-theory, we get H_{∗}^{V} (\underline{E}V → \{•\}; L_{R}^{⟨−∞⟩}) = 0.

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