On the classifying space of the family of finite and of virtually cyclic subgroups for CAT(0)-groups

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

February 4, 2009

Abstract

Let G be a discrete group which acts properly and isometrically on a complete CAT(0)-space X. Consider an integer d with d = 1 or d ≥ 3 such that the topological dimension of X is bounded by d. We show the existence of a G-CW -model \underline{E}G for the classifying space for proper Gactions with dim(\underline{E}G) ≤ d. Provided that the action is also cocompact, we prove the existence of a G-CW -model \underline{\underline{E}}G for the classifying space of the family of virtually cyclic subgroups satisfying dim(\underline{\underline{E}}G) ≤ d + 1.

Key words: classifying spaces of families, dimension, finite and virtually cyclic subgroups Mathematics Subject Classification 2000: 55R35.

0 Introduction

Given a group G, denote by \underline{E}G a G-CW -model for the classifying space for proper G-actions and by \underline{\underline{E}}G = E_{VCY}(G) a G-CW -model for the classifying space of the family of virtually cyclic subgroups. Our main theorem which will be proven in Section 3 is

Theorem 0.1. Let G be a discrete group which acts properly and isometrically on a complete proper CAT(0)-space X. Let top-dim(X) be the topological dimension of X. Let d be an integer satisfying d = 1 or d ≥ 3 such that top-dim(X) ≤ d.

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

(i) Then there is G-CW -model \underline{E}G with dim(\underline{E}G) ≤ d;

(ii) Suppose that G acts by semisimple isometries. (This is the case if we additionally assume that the G-action is cocompact.) Then there is G-CW -model \underline{\underline{E}}G with dim(\underline{\underline{E}}G) ≤ d + 1.

There is the question whether for any group G the inequality

hdim^{G}(\underline{E}G) − 1 ≤ hdim^{G}(\underline{\underline{E}}G) ≤ hdim^{G}(\underline{E}G) + 1  (0.2)

holds, where hdim^{G}(\underline{E}G) is the minimum of the dimensions of all possible G-CW -models for \underline{E}G and hdim^{G}(\underline{\underline{E}}G) is defined analogously (see [11, Introduction]). Since hdim(\underline{E}G) ≤ 1 + hdim(\underline{\underline{E}}G) holds for all groups G (see [11, Corollary 5.4]), Theorem 0.1 implies

Corollary 0.3. Let G be a discrete group and let X be complete CAT(0)-space X with finite topological dimension top-dim(X). Suppose that G acts properly and isometrically on X. Assume that the G-action is by semisimple isometries. (The last condition is automatically satisfied if we additionally assume that the G-action is cocompact.) Suppose that top-dim(X) = hdim^{G}(\underline{E}G) ≠ 2.

Then inequality (0.2) is true.

We will prove at the end of Section 3

Corollary 0.4. Suppose that G is virtually torsionfree. Let M be a simply connected complete Riemannian manifold of dimension n with non-negative sectional curvature. Suppose that G acts on M properly, isometrically and cocompactly. Then

hdim(\underline{E}G) = n;
n − 1 ≤ hdim(\underline{\underline{E}}G) ≤ n + 1.

In particular (0.2) holds.

If G is the fundamental group of an n-dimensional closed hyperbolic manifold, then hdim(\underline{E}G) = hdim(\underline{\underline{E}}G) = n by [11, Example 5.12]. If G is virtually ℤ^{n} for n ≥ 2, then hdim(\underline{E}G) = n and hdim(\underline{\underline{E}}G) = n+1 by [11, Example 5.21]. Hence the cases hdim(\underline{\underline{E}}G) = hdim(\underline{E}G) and hdim(\underline{\underline{E}}G) = hdim(\underline{E}G)+1 do occur in the situation of Corollary 0.4. There exists groups G with hdim(\underline{\underline{E}}G) = hdim(\underline{E}G) − 1 (see [11, Example 5.29]). But we do not believe that this is possible in the situation of Corollary 0.3 or Corollary 0.4.

The paper was supported by the Sonderforschungsbereich 478 – Geometrische Strukturen in der Mathematik – and the Max-Planck-Forschungspreis and the Leibniz-Preis of the author.

1 Classifying Spaces for Families

We briefly recall the notions of a family of subgroups and the associated classifying space. For more information, we refer for instance to the original source [13] or to the survey article [10].

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};
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 such that for any G-CW -complex Y with isotropy groups in F there exists a G-map Y → X and any two G-maps Y → X are G-homotopic. 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.

There exists a model for E_{F}(G) for any group G and any family F of subgroups. There is even a functorial construction (see [5, page 223 and Lemma 7.6 (ii)]).

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.

One can also define a numerable version of the space for proper G-actions to G which is denoted by \underline{J}G. It is not necessarily a G-CW -complex. A metric space X on which G acts isometrically and properly is a model for \underline{J}G if and only if the two projections X × X → X onto the first and second factor are G-homotopic to one another. If X is a complete CAT(0)-space on which G-acts properly and isometrically, then X is a model for \underline{J}G, the desired G-homotopy is constructed using the geodesics joining two points in X (see [3, Proposition 1.4 in II.1 on page 160]).

One motivation for studying the spaces \underline{E}G and \underline{\underline{E}}G comes from the Baum-Connes Conjecture and the Farrell-Jones Conjecture.

2 Topological and CW -dimension

Let X be a topological space. Let U be an open covering. Its dimension dim(U) ∈ {0, 1, 2, …} ⨿ {∞} is the infimum over all integers d ≥ 0 such that for any collection U_{0}, U_{1}, …, U_{d} of pairwise distinct elements in U the intersection ⋂_{i=0}^{d} U_{i} is empty. An open covering V is a refinement of U if for every V ∈ V there is U ∈ U with V ⊆ U.

Definition 2.1 (Topological dimension). The topological dimension of a topological space X

top-dim(X) ∈ {0, 1, 2, …} ⨿ {∞}

is the infimum over all integers d ≥ 0 such that any open covering U possesses a refinement V with dim(V) ≤ d.

Let Z be a metric space. We will denote for z ∈ Z and r ≥ 0 by B_{r}(z) and \overline{B}_{r}(z) respectively the open ball and closed ball respectively around z with radius r. We call Z proper if for each z ∈ Z and r ≥ 0 the closed ball \overline{B}_{r}(z) is compact. A group G acts properly on the topological space Z if for any z ∈ Z there is an open neighborhood U such that the set {g ∈ G | g · U ∩ U ≠ ∅} is finite. In particular every isotropy group is finite. If Z is a G-CW -complex, then Z is a proper G-space if and only if the isotropy group of any point in Z is finite (see [9, Theorem 1.23]).

Lemma 2.2. Let Z be a proper metric space. Suppose that G acts on Z isometrically and properly. Then we get for the topological dimensions of X and G\X

top-dim(G\X) ≤ top-dim(X).

Proof. Since G acts properly and isometrically, we can find for every z ∈ Z a real number ϵ(z) > 0 such that we have for all g ∈ G

g · B_{7ϵ(z)}(z) ∩ B_{7ϵ}(z) ≠ ∅ ⟺ g · B_{7ϵ(z)}(z) = B_{7ϵ(z)}(z) ⟺ g ∈ G_{z}.

We can arrange that ϵ(gz) = ϵ(z) holds for z ∈ Z and g ∈ G. Consider G · \overline{B}_{ϵ}(z). We claim that this set is closed in Z. We have to show for a sequence (z_{n})_{n≥0} of elements in \overline{B}_{ϵ}(z) and (g_{n})_{n≥0} of elements in G and x ∈ Z with lim_{n→∞} g_{n}z_{n} = x that x belongs to G · \overline{B}_{ϵ}(z). Since X is proper, we can find y ∈ \overline{B}_{ϵ}(z) such that lim_{n→∞} z_{n} = y. Choose N = N(ϵ) such that d_{X}(g_{n}z_{n}, x) ≤ ϵ and d_{X}(z_{n}, y) ≤ ϵ holds for n ≥ N. We conclude for n ≥ N

d_{x}(g_{n}y, x) ≤ d_{X}(g_{n}y, g_{n}z_{n}) + d_{X}(g_{n}z_{n}, x)
= d_{X}(y, z_{n}) + d_{X}(g_{n}z_{n}, x)
≤ ϵ + ϵ
= 2ϵ.

This implies for n ≥ N

d_{X}(g_{n}^{−1}g_{N}z, z) = d_{X}(g_{N}z, g_{n}z)
≤ d_{X}(g_{N}z, g_{N}y) + d_{X}(g_{N}y, x) + d_{X}(x, g_{n}y) + d_{X}(g_{n}y, g_{n}z)
= d_{X}(z, y) + d_{X}(g_{N}y, x) + d_{X}(g_{n}y, x) + d_{X}(y, z)
≤ ϵ + 2ϵ + 2ϵ + ϵ
= 6ϵ.

Hence g_{n}^{−1}g_{N} ∈ G_{z} for n ≥ N. Since G_{z} is finite, we can arrange by passing to subsequences that g_{0} = g_{n} holds for n ≥ 0. Hence

x = lim_{n→∞} g_{n}z_{n} = lim_{n→∞} g_{0}z_{n} = g_{0} · lim_{n→∞} z_{n} = g_{0} · y ∈ G · \overline{B}_{ϵ}(z).

Choose a set-theoretic section s: G/G_{z} → G of the projection G → G/G_{z}. The map

G/G_{z} × B_{7ϵ(z)}(z) ⟶^{≅} G · B_{7ϵ(z)}(z), (gG_{z}, x) ↦ s(gG_{z}) · x

is bijective, continuous and open and hence a homeomorphism. It induces a homeomorphism

G/G_{z} × \overline{B}_{ϵ(z)}(z) ⟶^{≅} G · \overline{B}_{ϵ(z)}(z).

This implies

top-dim(\overline{B}_{ϵ(z)}(z)) = top-dim(G · \overline{B}_{ϵ(z)}(z)).  (2.3)

Let pr: Z → G\Z be the projection. It induces a bijective continuous map G_{z}\\overline{B}_{ϵ(z)}(z) ⟶^{≅} pr(\overline{B}_{ϵ(z)}(z)) which is a homeomorphism since \overline{B}_{ϵ(z)}(z) and hence G_{z}\\overline{B}_{ϵ(z)}(z) is compact. Hence we get

top-dim(pr(\overline{B}_{ϵ(z)}(z))) = top-dim(G_{z}\\overline{B}_{ϵ(z)}(z)).  (2.4)

Since the metric space \overline{B}_{ϵ(z)}(z) is compact and hence contains a countable dense set and G_{z} is finite, we conclude from [2, Exercise in Chapter II on page 112]

top-dim(G_{z}\\overline{B}_{ϵ(z)}(z)) ≤ top-dim(\overline{B}_{ϵ(z)}(z)).  (2.5)

From (2.3), (2.4) and (2.5) we conclude that G·\overline{B}_{ϵ(z)}(z) ⊆ Z and pr(\overline{B}_{ϵ(z)}(z)) ⊆ G\Z are closed and satisfy

top-dim(pr(\overline{B}_{ϵ(z)}(z))) ≤ top-dim(G · \overline{B}_{ϵ(z)}(z)).  (2.6)

Since Z is proper, it is the countable union of compact subspaces and hence contains a countable dense subset. This is equivalent to the condition that Z has a countable basis for its topology. Obviously the same is true for G\Z. We conclude from [12, Theorem 9.1 in in Chapter 7.9 on page 302 and Exercise 9 in Chapter 7.9 on page 315]

top-dim(Z) = sup\{top-dim(G · \overline{B}_{ϵ(z)}(z))\};  (2.7)
top-dim(G\Z) = sup\{top-dim(pr(\overline{B}_{ϵ(z)}(z)))\}.  (2.8)

Now Lemma 2.2 follows from (2.6), (2.7) and (2.8).

In the sequel we will equip a simplicial complex with the weak topology, i.e., a subset is closed if and only if its intersection with any simplex σ is a closed subset of σ. With this topology a simplicial complex carries a canonical CW -structure.

Let X be a G-space. We call a subset U ⊆ X a FIN -set if we have gU ∩ U ≠ ∅ ⟹ gU = U for every g ∈ G and G_{U} := {g ∈ G | g · U = U} is finite. Let U be a covering of X by open FIN -subset. Suppose that U is G-invariant, i.e., we have g · U ∈ U for g ∈ G and U ∈ U. Define its nerve N (U) to be the simplicial complex whose vertices are the elements in U and for which the pairwise distinct vertices U_{0}, U_{1}, …, U_{d} span a d-simplex if and only if ⋂_{i=0}^{d} U_{i} ≠ ∅. The action of G on X induces an action on U and hence a simplicial action on N (U). The isotropy group of any vertex is finite and hence the isotropy group of any simplex is finite. Let N (U)′ be the barycentric subdivision. It inherits a simplicial G-action from N (U) such that for any g ∈ G and any simplex σ whose interior is denoted by σ^{∘} and which satisfies g · σ^{∘} ∩ σ^{∘} ≠ ∅ we have gx = x for all x ∈ σ^{∘}. In particular N (U)′ is a G-CW -complex and agrees as a G-space with N (U).

Lemma 2.9. Let n be an integer with n ≥ 0. Let X be a proper metric space whose topological dimension satisfies top-dim(X) ≤ n. Suppose that G acts properly and isometrically on X.

Then there exists a proper n-dimensional G-CW -complex Y together with a G-map f: X → Y.

Proof. Since the G-action is proper we can find for every x ∈ X an ϵ(x) > 0 such that for every g ∈ G we have

g · \overline{B}_{2ϵ(x)}(x) ∩ \overline{B}_{2ϵ(x)}(x) ≠ ∅ ⇔ g · \overline{B}_{2ϵ(x)}(x) = \overline{B}_{2ϵ(x)}(x) ⇔ g · B_{2ϵ(x)}(x) = B_{2ϵ(x)}(x) ⇔ g · B_{ϵ(x)}(x) = B_{ϵ(x)}(x) ⇔ g ∈ G_{x}.

We can arrange that ϵ(gx) = ϵ(x) for g ∈ G and x ∈ X holds. We obtain a covering of X by open FIN -subsets {B_{ϵ(x)}(x) | x ∈ X}. Let pr: X → G\X be the canonical projection. We obtain an open covering of G\X by {pr(B_{ϵ(x)}(x)) | x ∈ X}. Since top-dim(X) ≤ n by assumption and G acts properly on X, we get top-dim(G\X) ≤ n from Lemma 2.2. Since G acts properly and isometrically on X, the quotient G\X inherits a metric from X. Hence G\X is paracompact by Stone’s theorem (see [12, Theorem 4.3 in Chap. 6.3 on page 256]) and in particular normal. By [6, Theorem 3.5 on page 211] we can find a locally finite open covering U of G\X such that dim(U) ≤ n and U is a refinement of {pr(B_{ϵ(x)}(x)) | x ∈ X}. For each U ∈ U choose x(U) ∈ X with U ⊆ pr(B_{ϵ(U)}(x(U)). Define the index set

J = \{(U, \overline{g}) | U ∈ U, \overline{g} ∈ G/G_{x(U)}\}.

For (U, \overline{g}) ∈ J define an open FIN -subset of X by

V_{U,\overline{g}} := pr^{−1}(U) ∩ g · B_{2ϵ(x(U))}(x(U)).

Obviously this is well-defined, i.e., the choice of g ∈ \overline{g} does not matter, and we have pr(V_{U,\overline{g}}) ⊆ U and V_{U,\overline{g}} ⊆ g · B_{2ϵ(x(U))}(x(U)).

Consider the collection of subsets of X

V = \{V_{U,\overline{g}} | (U, \overline{g}) ∈ J\}.

This is a G-invariant covering of X by open FIN -subsets. Its dimension satisfies

dim(V) ≤ dim(U) ≤ n

since for U ∈ U, \overline{g_{1}}, \overline{g_{2}} ∈ G/G_{x(U)} we have

V_{U,\overline{g_{1}}} ∩ V_{U,\overline{g_{2}}} ≠ ∅ ⟹ g_{1} · B_{2ϵ(x(U))}(x(U)) ∩ g_{2} · B_{2ϵ(x(U))}(x(U)) ⟹ \overline{g_{1}} = \overline{g_{2}}.

Since U is locally finite and G\X is paracompact, we can find a locally finite partition of unity {e_{U}: G\X → [0, 1] | U ∈ U} which is subordinate to U, i.e., ∑_{U∈U} e_{U} = 1 and supp(e_{U}) ⊂ U for every U ∈ U. Fix a map χ: [0, ∞) → [0, 1] satisfying χ^{−1}(0) = [1, ∞). Define for (U, \overline{g}) ∈ J a function

ϕ_{U,\overline{g}}: X → [0, 1], y ↦ e_{U}(pr(y)) · χ(d_{X}(y, gx(U))/ϵ(x(U))).

Consider y ∈ X. Since U is locally finite and G\X is locally compact, we can find an open neighborhood T of pr(y) such that \overline{T} meets only finitely many elements of U. Choose an open neighborhood W_{0} of y such that \overline{W_{0}} is compact. Define an open neighborhood of y by

W := W_{0} ∩ pr^{−1}(T).

Since \overline{W_{0}} is compact, \overline{W} is compact. Since G acts properly, there exists for a given U ∈ U only finitely many elements g ∈ G with \overline{W} ∩ g · B_{ϵ(x(U))}(x(U)) ≠ ∅. Since \overline{T} meets only finitely elements of U, the set

J_{W} := \{(U, \overline{g}) ∈ J | \overline{W} ∩ g · B_{ϵ(x(U))}(x(U)) ∩ pr^{−1}(U) ≠ ∅\}

is finite. Suppose ϕ_{U,\overline{g}}(z) > 0 for (U, \overline{g}) ∈ J and z ∈ W. We conclude z ∈ pr^{−1}(U) ∩ g · B_{ϵ(x(U))}(x(U)) and hence (U, \overline{g}) ∈ J_{W}. Thus we have shown that the collection \{ϕ_{U,\overline{g}} | (U, \overline{g}) ∈ J\} is locally finite.

We conclude that the map

∑_{(U,\overline{g})∈J} ϕ_{U,\overline{g}}: X → [0, 1], y ↦ ∑_{(U,\overline{g})∈J} e_{U}(pr(y)) · χ(d_{X}(y, gx(U))/ϵ(x(U)))

is well-defined and continuous. It has always a value greater than zero since for every y ∈ X there exists U ∈ U with e_{U}(pr(y)) > 0, the set pr^{−1}(U) is contained in ⋃_{g∈G} g · B_{ϵ(U)}(x(U)) and χ^{−1}(0) = [1, ∞). Define for (U, \overline{g}) ∈ J a map

ψ_{U,\overline{g}}: X → [0, 1], y ↦ \frac{ϕ_{U,\overline{g}}(y)}{∑_{(U,\overline{g})∈J} ϕ_{U,\overline{g}}(y)}.

We conclude that

∑_{(U,\overline{g})∈J} ψ_{U,\overline{g}}(y) = 1  for y ∈ X;
ψ_{U,\overline{g}}(hy)  = ψ_{U,\overline{h^{−1}g}}(y)  for h ∈ G, y ∈ Y and (U, \overline{g}) ∈ J;
supp(ψ_{U,\overline{g}})  ⊆ V_{U,\overline{g}}  for (U, \overline{g}) ∈ J,

and the collection \{ψ_{U,\overline{g}} | (U, \overline{g}) ∈ J\} is locally finite. Define the desired proper n-dimensional G-CW -complex to be the nerve Y := N (V). Define a map by

f: X → N (V), y ↦ ∑_{(U,\overline{g})∈J} ψ_{U,\overline{g}}(y) · V_{U,\overline{g}}.

It is well-defined since for y ∈ X the simplices V_{U,\overline{g}} for which ψ_{U,\overline{g}}(y) ≠ 0 holds span a simplex because y ∈ X with ψ_{U,\overline{g}}(y) ≠ 0 belongs to V_{U,\overline{g}} and hence the intersection of the sets V_{U,\overline{g}} for which ψ_{U,\overline{g}}(y) ≠ 0 holds contains y and hence is non-empty. The map f is continuous since \{ψ_{U,\overline{g}} | (U, \overline{g}) ∈ J\} is locally finite. It is G-equivariant by the following calculation for h ∈ G and y ∈ Y:

f(hy) = ∑_{(U,\overline{g})∈J} ψ_{U,\overline{g}}(hy) · V_{U,\overline{g}}
= ∑_{(U,\overline{g})∈J} ψ_{U,\overline{hg}}(hy) · V_{U,\overline{hg}}
= ∑_{(U,\overline{g})∈J} ψ_{U,\overline{h^{−1}hg}}(y) · V_{U,\overline{hg}}
= ∑_{(U,\overline{g})∈J} ψ_{U,\overline{g}}(y) · h · V_{U,\overline{g}}
= h · ∑_{(U,\overline{g})∈J} ψ_{U,\overline{g}}(y) · V_{U,\overline{g}}
= h · f(y).

Lemma 2.10. Let X and Y be G-CW -complexes. Let i: X → Y and r: Y → X be G-maps such that r ∘ i is G-homotopic to the identity. Consider an integer d ≥ 3. Suppose that Y has dimension ≤ d.

Then X is G-homotopy equivalent to a G-CW -complex Z of dimension ≤ d.

Proof. By the Equivariant Cellular Approximation Theorem (see [14, Theorem II.2.1 on page 104]) we can assume without loss of generality that i and r are cellular. Let cyl(r) be the mapping cylinder. Let k: Y → cyl(r) be the canonical inclusion and p: cyl(r) → X be the canonical projection. Then p is a G-homotopy equivalence and p ∘ k = r. Let Z be the union of the 2-skeleton of cyl(r) and Y. This is a G-CW -subcomplex of cyl(r) and cyl(r) is obtained from Z by attaching equivariant cells of dimension ≥ 3. Hence the map p|_{Z}: Z → X has the property that it induces on every fixed point set a 2-connected map. Let j: X → Z be the composite of i: X → Y with the obvious inclusion Y → Z. Then p|_{Z} ∘ j = p ∘ k ∘ i = r ∘ i is G-homotopy equivalent to the identity and the dimension of Z is still bounded by d since we assume d ≥ 3. Hence we can assume in the sequel that r^{H}: Y^{H} → X^{H} is 2-connected for all H ⊆ G, otherwise replace Y by Z, i by j and r by p|_{Z}.

We want to apply [9, Proposition 14.9 on page 282]. Here the assumption d ≥ 3 enters. Hence it suffices to show that the cellular ℤΠ(G, X)-chain complex C_{∗}^{c}(X) is ℤΠ(G, X)-chain homotopy equivalent to a d-dimensional ℤΠ(G, X)- chain complex. By [9, Proposition 11.10 on page 221] it suffices to show that the cellular ℤΠ(G, X)-chain complex C_{∗}^{c}(X) is dominated by a d-dimensional ℤΠ(G, X)-chain complex. This follows from the geometric domination (Y, i, r) by passing to the cellular chain complexes over the fundamental categories since r and hence also i induce equivalences between the fundamental categories because r^{H}: Y^{H} → X^{H} is 2-connected for all H ⊆ G and r ∘ i ≃_{G} id_{X}.

The condition d ≥ 3 is needed since we want to argue first with the cellular ℤOr(G)-chain complex and then transfer the statement that it is d-dimensional to the statement that the underlying G-CW -complex is d-dimensional. The condition d ≥ 3 enters for analogous reasons in the classical proof of the theorem that the existence of a d-dimensional ℤG-projective resolution for the trivial ℤG-module ℤ implies the existence of a d-dimensional model for BG (see [4, Theorem 7.1 in Chapter VIII.7 on page 205]).

Theorem 2.11. Let G be a discrete group. Then

(i) There is a G-homotopy equivalence \underline{J}G → \underline{E}G;

(ii) Suppose that there is a model for \underline{J}G which is a metric space such that the action of G on \underline{J}G is isometric. Consider an integer d with d = 1 or d ≥ 3. Suppose that the topological dimension top-dim(\underline{J}G) ≤ d.

Then there is a G-CW -model for \underline{E}G of dimension ≤ d;

(iii) Let d be an integer d ≥ 0. Suppose that there is a G-CW -model for \underline{E}G with dim(\underline{E}G) ≤ d such that \underline{E}G after forgetting the group action has countably many cells.

Then there exists a model for \underline{J}G with top-dim(\underline{J}G) ≤ d.

Proof. (i) This is proved in [10, Lemma 3.3 on page 278]. (ii) Choose a G-homotopy equivalence i: \underline{E}G → \underline{J}G. From Lemma 2.9 we obtain a G-map f: \underline{J}G → Y to a proper G-CW -complex of dimension ≤ d. By the universal properly of \underline{E}G we can find a G-map h: Y → \underline{E}G and the composite h ∘ f ∘ i is G-homotopic to the identity on \underline{E}G.

Suppose d ≥ 3. We conclude from Lemma 2.10 that \underline{E}G is G-homotopy equivalent to a G-CW -complex of dimension ≤ d.

Suppose d = 1. By Dunwoody [7, Theorem 1.1] it suffices to show that the rational cohomological dimension of G satisfies cd_{ℚ}(G) ≤ 1. Hence we have to show for any ℚG-module M that Ext_{ℚG}^{n}(ℚ, M) = 0 for n ≥ 2, where ℚ is the trivial ℚG-module. Since all isotropy groups of \underline{E}G and Y are finite, their cellular ℚG-chain complexes are projective. Since \underline{E}G is contractible, C_{∗}(\underline{E}G; ℚ) is a projective ℚG-resolution and hence

Ext_{ℚG}^{n}(ℚ, M) ≅ H^{n}(hom_{ℚG}(C_{∗}(\underline{E}G; ℚ), M)).

Since h ∘ f ∘ i ≃_{G} id_{\underline{E}G}, the ℚ-module H^{n}(hom_{ℚG}(C_{∗}(\underline{E}G; ℚ), M)) is a direct summand in the ℚ-module H^{n}(hom_{ℚG}(C_{∗}(Y; ℚ), M)). Since Y is 1-dimensional by assumption, H^{n}(hom_{ℚG}(C_{∗}(Y; ℚ), M)) vanishes for n ≥ 2. This implies that Ext_{ℚG}^{n}(ℚ, M) vanishes for n ≥ 2. (iii) Using the equivariant version of the simplicial approximation theorem and the fact that changing the G-homotopy class of attaching maps does not change the G-homotopy type, one can find a simplicial complex X with simplicial Gaction which is G-homotopy equivalent to \underline{E}G, satisfies dim(X) = dim(\underline{E}G) and has only countably many simplices. Hence the barycentric subdivision X′ is a simplicial complex of dimension ≤ d with countably many simplices and carries a G-CW -structure. The latter implies that X′ is a G-CW -model for \underline{E}G and hence also a model for \underline{J}G. Since the dimension of a simplicial complex with countably many simplices is equal to its topological dimension, we conclude top-dim(X′) = dim(X) = dim(\underline{E}G) ≤ d.

3 The passage from finite to virtually cyclic groups

In [11] it is described how one can construct \underline{\underline{E}}G from \underline{E}G. In this section we want to make this description more explicit under the following condition

Condition 3.1. We say that G satisfies condition (C) if for every g, h ∈ G with |h| = ∞ and k, l ∈ ℤ we have

gh^{k}g^{−1} = h^{l} ⟹ |k| = |l|.

Let ICY be the set of infinite cyclic subgroup C of G. This is not a family since it does not contain the trivial subgroup. We call C, D ∈ ICY equivalent if |C ∩ D| = ∞. One easily checks that this is an equivalence relation on ICY. Denote by [ICY] the set of equivalence classes and for C ∈ ICY by [C] its equivalence class. Denote by

N_{G}C := {g ∈ G | gCg^{−1} = C}

the normalizer of C in G. Define for [C] ∈ [ICY] a subgroup of G by

N_{G}[C] := {g ∈ G | |gCg^{−1} ∩ C| = ∞}.

One easily checks that this is independent of the choice of C ∈ [C]. Actually N_{G}[C] is the isotropy of [C] under the action of G induced on [ICY] by the conjugation action of G on ICY.

Lemma 3.2. Suppose that G satisfies Condition (C) (see 3.1). Consider C ∈ ICY.

Then obtain a nested sequence of subgroups

N_{G}C ⊆ N_{G}2!C ⊆ N_{G}3!C ⊆ N_{G}4!C ⊆ ⋯

where k!C is the subgroup of C given by {h^{k!} | h ∈ C}, and we have

N_{G}[C] = ⋃_{k≥1} N_{G}k!C.

Proof. Since every subgroup of a cyclic group is characteristic, we obtain the nested sequence of normalizers N_{G}C ⊆ N_{G}2!C ⊆ N_{G}3!C ⊆ N_{G}4!C ⊆ · · ·.

Consider g ∈ N_{G}[C]. Let h be a generator of C. Then there are k, l ∈ ℤ with gh^{k}g^{−1} = h^{l} and k, l ≠ 0. Condition (C) implies k = ±l. Hence g ∈ N_{G}⟨h^{k}⟩ ⊆ N_{G}k!C. This implies N_{G}[C] ⊆ ⋃_{k≥1} N_{G}k!C. The other inclusion follows from the fact that for g ∈ N_{G}k!C we have k!C ⊆ gCg^{−1} ∩ C.

Fix C ∈ ICY. Define a family of subgroups of N_{G}[C] by

G_{G}(C) := {H ⊆ N_{G}[C] | [H : (H ∩ C)] < ∞}

∪ {H ⊆ N_{G}[C] | |H| < ∞}. (3.3)

Notice that G_{G}(C) consists of all finite subgroups of N_{G}[C] and of all virtually cyclic subgroups of N_{G}[C] which have an infinite intersection with C. Define a quotient group of N_{G}C by

W_{G}C := N_{G}C/C.

Lemma 3.4. Let n be an integer. Suppose that G satisfies Condition (C) (see 3.1). Suppose that there exists a G-CW -model for \underline{E}G with dim(\underline{E}G) ≤ n and for every C ∈ ICY there exists a W_{G}C-CW -model for \underline{E}W_{G}C with dim(\underline{E}W_{G}C) ≤ n.

Then there exists a G-CW -model for \underline{\underline{E}}G with dim(\underline{\underline{E}}G) ≤ n + 1.

Proof. Because of [11, Theorem 2.3 and Remark 2.5] it suffices to show for every C ∈ ICY that there is a N_{G}[C]-model for E_{G_{G}(C)}(N_{G}[C]) with

dim(E_{G_{G}(C)}(N_{G}[C])) ≤ n + 1.  (3.5)

Because of Lemma 3.2 we have

N_{G}[C] = colim_{k→∞} N_{G}k!C.

We conclude (3.5) from [11, Lemma 4.2 and Theorem 4.3] since every element H ∈ G_{G}(C) is finitely generated and hence lies already in N_{G}k!C for some k > 0, by assumption there exists a W_{G}k!C-CW -model for \underline{E}W_{G}k!C with dim(\underline{E}W_{G}k!C) ≤ n, and res_{N_{G}k!C→W_{G}k!C} \underline{E}W_{G}k!C is E_{G_{G}(C)|_{N_{G}k!C)}}(N_{G}k!C).

Now we are ready to prove Theorem 0.1.

Proof of Theorem 0.1. (i) Consider an integer d ∈ ℤ with d = 1 or d ≥ 3 such that d ≥ top-dim(X). The space X is a model for \underline{J}G by [3, Corollary 2.8 in II.2. on page 178]. We conclude from Theorem 2.11 (ii) that there is a d-dimensional model for \underline{E}G.

(ii) We will use in the proof some basic facts and notions about isometries of proper complete CAT(0)-spaces which can be found in [3, Chapter II.6].

The group G satisfies condition (C) by the following argument. Suppose that gh^{k}g^{−1} = h^{l} for g, h ∈ G with |h| = ∞ and k, l ∈ ℤ. The isometry l_{h}: X → X given by multiplication with h is a hyperbolic isometry since it has no fixed point and is by assumption semisimple. We obtain for the translation length L(h) which is a real number satisfying L(h) > 0

k · L(h) = L(h^{k}) = L(gh^{k}g^{−1}) = L(h^{l}) = l · L(h).

This implies k = l.

Let C ⊆ G be any infinite cyclic subgroup. Choose a generator g ∈ C. The isometry l_{g}: X → X given by multiplication with g is a hyperbolic isometry. Let Min(g) ⊂ X be the the union of all axes of g. Then Min(g) is a closed convex subset of X. There exists a closed convex subset Y (g) ⊆ X and an isometry

α: Min(g) ⟶^{≅} Y (g) × ℝ.

The space Min(G) is N_{G}C-invariant since for each h ∈ N_{G}C we have hgh^{−1} = g or hgh^{−1} = g^{−1} and hence multiplication with h sends an axis of g to an axis of g. The N_{G}C-action induces a proper isometric W_{G}C-action on Y (g). These claims follow from [3, Theorem 6.8 in II.6 on page 231 and Proposition 6.10 in II.6 on page 233]. The space Y (g) inherits from X the structure of a CAT(0)-space and satisfies top-dim(Y (g)) ≤ top-dim(X). Hence Y (g) is a model for \underline{J}W_{G}C with top-dim(Y (g)) ≤ top-dim(X) by [3, Corollary 2.8 in II.2. on page 178]. We conclude from Theorem 2.11 (ii) that there is a d-dimensional model for \underline{E}W_{G}C for every infinite cyclic subgroup C ⊆ G. Now Theorem 0.1 follows from Lemma 3.4.

Finally we prove Corollary 0.4.

Proof of Corollary 0.4. A complete Riemannian manifold M with non-negative sectional curvature is a CAT(0)-space (see [3, Theorem IA.6 on page 173 and Theorem II.4.1 on page 193].) Since G is virtually torsionfree, we can find a subgroup G_{0} of finite index in G such that G_{0} is torsionfree and acts orientation preserving on M. Hence G_{0}\M is a closed orientable manifold of dimension n. Hence H_{n}(M; ℤ) = H_{n}(BG; ℤ) ≠ 0. This implies that every CW -model BG_{0} has at least dimension n. Since the restriction of \underline{E}G to G_{0} is a G_{0}-CW -model for EG_{0}, we conclude hdim(\underline{E}G) ≥ n. Since M with the given G_{0}-action is a G-CW -model for \underline{E}G (see [1, Theorem 4.15]), we conclude

hdim(\underline{E}G) = n = top-dim(M).

If n ≠ 2, we conclude hdim(\underline{\underline{E}}G) ≤ n + 1 from Theorem 0.1. Since hdim(\underline{E}G) ≤ 1 + hdim(\underline{\underline{E}}G) holds for all groups G (see [11, Corollary 5.4]), we get

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

provided that n ≠ 2.

Suppose n = 2. If G_{0} is a torsionfree subgroup of finite index in G, then G_{0}\X is a closed 2-dimensional manifold with non-negative sectional curvature. Hence G_{0} is ℤ^{2} or hyperbolic. This implies that G is virtually ℤ^{2} or hyperbolic. Hence hdim(\underline{\underline{E}}G) ∈ \{2, 3\} by [11, Example 5.21] in the first case and by [11, Theorem 3.1, Example 3.6, Theorem 5.8 (ii)] or [8, Proposition 6, Remark 7 and Proposition 8] in the second case.

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