London Mathematical Society ISSN 1461–1570

ON GROUPS OF TYPE Φ

OLYMPIA TALELLI

Abstract

The classifying space \underline{E}G for the family of the finite subgroups of G is defined up to G-homotopy as a G-CW-complex \underline{E}G such that \underline{E}G^{H} is contractible if H is finite and empty otherwise.

We define a group G to be of type Φ if it has the property that for every ℤG-module M, proj. dim_{ℤG}M < ∞ iff proj. dim_{ℤH}M < ∞ for every finite subgroup H of G.

We show that if N_{G}(H)/H is of type Φ for every finite subgroup H of G and dim | Λ(G) |< ∞, where | Λ(G) | is the G-simplicial complex determined by the poset of the nontrivial finite subgroups of G, then G admits a finite dimensional model for \underline{E}G.

This generalizes a theorem of P. Kropholler and G. Mislin [6] and a theorem of W. Lück [8].

We also propose the type Φ as an algebraic characterization of those groups which admit a finite dimensional model for \underline{E}G.

1. Introduction

Associated to any group G there is a G-CW -complex \underline{E}G with the following properties.

Every cell stabilizer is finite and for each finite subgroup H of G, the fixed point subcomplex \underline{E}G^{H} is contractible. Moreover, for any G-CW -complex X with finite cell stabilizers there is precisely

one G-map up to G-homotopy from X to \underline{E}G and thus any two models of an \underline{E}G are G-homotopy equivalent.

In [6] P. Kropholler and G. Mislin proved Theorem A. Every HF-group G of type F P_{∞} admits a finite dimensional model

for \underline{E}G.

The class HF was introduced by P. Kropholler in [5] and it is defined as the smallest class of groups containing the class of finite groups with the property: if a group G admits a finite dimensional contractible G-CW -complex with all cell stabilizers in HF, then G is in HF.

To prove Theorem A they first considered the following two properties of HFgroups of type F P_{∞}, which were both shown by using complete cohomology.

• if G is an HF-group of type F P_{∞} then there is a bound on the orders of the finite subgroups of G.

Received ???, revised ???. 2000 Mathematics Subject Classification 20J05, 55N25 \begin{matrix}c \\ ◯ \end{matrix} ????, Olympia Talelli

LMS J. Comput. Math. ?? (????) 1–8

On groups of type Φ

In particular, if | Λ(G) | is the G-simplicial complex determined by the poset of the non-trivial finite subgroups of G, then dim | Λ(G) |< ∞.

• If G is an HF-group of type F P_{∞} then proj. dim_{ℤG}B(G, ℤ) < ∞, where B(G, ℤ) is the ℤG-module of bounded functions from G to ℤ.

They then proved.

Theorem B. If G is an HF-group such that dim | Λ(G) |< ∞ and proj. dim_{ℤG} B(G, ℤ) < ∞ then there is a finite dimensional model for \underline{E}G. Clearly now Theorem A is an immediate consequence of Theorem B. A crucial ingredient of the proof of Theorem B was the following result of Cornick and Kropholler [2]: if G is an HF-group with proj. dim_{ℤG}B(G, ℤ) < ∞ and M is a ℤG-module, which is projective as a ℤH-module for all finite subgroups H of G, then proj. dim_{ℤG}M < ∞.

The dimension estimate for \underline{E}G that they obtained in [6] is exponential in dim | Λ(G) |. In [8] W. Lück proved

Theorem C. If G is a group such that dim | Λ(G) |< ∞ and G satisfies B(n), for some non-negative integer n, then G admits a model for \underline{E}G of dimension at most max{3, n} + λ(n + 1). W. Lück defines a group G to satisfy condition b(n), for some non-negative integer n, if every ℤG-module M, which is projective as a ℤH-module for all finite subgroups H of G, has proj. dim_{ℤG}M ⩽ n. He then defines a group G to satisfy condition B(n), for some non-negagive integer n, if the group N_{G}(H)/H satisfies b(n) for every finite subgroup H of G. The non-negative integer λ is defined as a bound on the lengths l(H) of the finite subgroups H of G, where

l(H) = sup{H_{0} < H_{2} < ⋯ < H_{τ} : H_{i} ⩽ H, H_{i} ≠ H_{i+1}, i = 0, …, τ}.

We define a group G to be of type Φ if it has the property that for every ℤGmodule M, proj. dim_{ℤG}M < ∞ if and only if proj. dim_{ℤH}M < ∞ for every finite subgroup H of G. We show

Theorem 1.1. If G is a group such that N_{G}(H)/H is of type Φ for every finite subgroup H of G and dim | Λ(G) |< ∞ then G admits a finite dimensional model for \underline{E}G.

The proof of this theorem follows essentially the proof of Theorem B of Kropholler and Mislin in [6].

Note that if G is an HF-group then G is of type Φ if and only if proj. dim_{ℤG} B(G, ℤ) < ∞. Moreover, if G is an HF-group and H is a finite subgroup of G then N_{G}(H)/H is an HF-group.

We show that if N_{G}(H)/H is of type Φ for every finite subgroup H of G, then G satisfies condition B(n_{0}), where n_{0} = proj. dim_{ℤG}B(G, ℤ).

Thus it follows from W. Lück’s Theorem C that G admits a model for \underline{E}G of dimension at most max{3, n_{0}} + λ(n_{0} + 1).

If G admits a finite dimensional \underline{E}G then N_{G}(H)/H is of type Φ for every finite subgroup H of G. However, the condition dim | Λ(G) |< ∞ is not a necessary condition for G to admit a finite dimensional model for \underline{E}G.

We propose the following as an algebraic characterization for those groups G which admit a finite dimensional model for \underline{E}G.

Conjecture A. The following statements are equivalent for a group G

(1) G admits a finite dimensional model for \underline{E}G

(2) G is of type Φ

(3) spliℤG < ∞

(4) silpℤG < ∞

(5) fin. dim ℤG < ∞. The algebraic invariants spliℤG and silpℤG were defined in [3], silpℤG is the supremum of the injective lengths of the projective ℤG-modules and spliℤG is the supremum of the projective lengths of the injective ℤG-modules. It was shown in [3] that silpℤG ⩽ spliℤG and that if spliℤG < ∞ then silpℤG = spliℤG.

The finitistic dimension of ℤG, fin. dim ℤG, is the supremum of the projective dimensions of the ℤG-modules of finite projective dimension.

We show (Prop. 3.3, Prop. 2.3, Prop. 2.2)

• (1) ⇒ (2) ⇒ (3) ⇒ (4) ⇒ (5)

• (2) ⇒ fin. dim ℤG = silpℤG = spliℤG = m and m − 1 ⩽ proj. dim_{ℤG}B(G, ℤ) ⩽ m.

Note that it follows from Theorem B of Kropholler and Mislin that 5) ⇒ 1) if G is an HF–group with dim |Λ(G)| < ∞.

We show (Cor. 3.6), that (5) ⇒ (1), if G is a torsion-free locally soluble group.

2. Algebraic Invariants

Proposition 2.1. If proj. dim_{ℤG}B(G, ℤ) < ∞ and M is a ℤ-free ℤG-module of finite projective dimension then proj. dim_{ℤG}M ⩽ proj. dim_{ℤG}B(G, ℤ).

Proof. We know that B(G, ℤ) is ℤ-free and that there is a ℤG-exact sequence 0 → ℤ → B(G, ℤ) → \overline{B} → 0 which is ℤ-split [6]. So in particular, \overline{B} is ℤ-free. If proj. dim_{ℤG}N = k and 0 → P_{k} → ⋯ → P_{0} → N → 0 is a projective resolution

of N then for any ℤ-free ℤG-module A, 0 → P^{↘}_{k} ⊗ A^{↘} → ⋯ → P^{↘}_{0} ⊗ A^{↘} → N^{↘} ⊗ A^{↘} → 0

is a projective resolution for N^{↘} ⊗ A^{↘}. Hence if n = proj. dim_{ℤG}B(G, ℤ) and m = proj. dim_{ℤG}M then proj. dim_{ℤG}M ⊗ B(G, ℤ) ⩽ m, n and proj. dim_{ℤG}M ⊗ \overline{B} ⩽ m. It now follows from the long exact Ext-sequence associated to

0 → M → M ⊗ B(G, ℤ) → M ⊗ \overline{B} → 0

that if proj. dim_{ℤG}M > n then proj. dim_{ℤG}M ⊗ \overline{B} ⩾ m + 1, a contradiction, hence proj. dim_{ℤG}M ⩽ n.

In [2] it was shown that if G is an HF-group then

silpG = spliG = k(ℤG) = fin. dim ℤG

where

k(ℤG) = sup{proj. dim_{ℤG}M | proj. dim_{ℤG}M < ∞ for every finite subgroup H ⩽ G}.

We show

Proposition 2.2. Let G be any group. Then

fin. dim ℤG ⩽ silpℤG ⩽ spliℤG ⩽ k(ℤG).

Moreover, if k(ℤG) < ∞ and k(ℤG) = n then

fin. dim ℤG = silpℤG = spliℤG = k(ℤG)

and

n − 1 ⩽ proj. dim_{ℤG}B(G, ℤ) ⩽ n.

Proof. It is easy to see that fin. dim ℤG ⩽ silpℤG and in [3] it was shown that silpℤG ⩽ spliℤG. Let H be a finite subgroup of G and I an injective ℤG-module. Then I is an injective ℤH-module, hence I is cohomologically trivial which implies that proj. dim_{ℤH}I < ∞. Thus spliℤG ⩽ k(ℤG). If k(ℤG) < ∞ then clearly k(ℤG) ⩽ fin. dim ℤG hence fin. dim ℤG = silpℤG = spliℤG = k(ℤG). If k(ℤG) = n then as B(G, ℤ) is a free ℤH-module for every finite subgroup H of G [7], it follows that proj. dim_{ℤG}B(G, ℤ) ⩽ n. It follows easily now from Proposition 2.1 that n − 1 ⩽ proj. dim_{ℤG}B(G, ℤ) ⩽ n.

Note that in this case proj. dim_{ℤG}B(G, ℤ) = sup{k : Ext_{ℤG}^{k}(M, F) ≠ 0, M ℤ-free, F ℤG-free}. The last invariant was introduced and studied by B. Ikenaga in [4].

Proposition 2.3. The following statements are equivalent for any group G

(i) G is of type Φ (ii) proj. dim_{ℤG}B(G, ℤ) < ∞ and G satisfies b(n_{0}), where n_{0} = proj. dim_{ℤG}B(G, ℤ) (iii) k(ℤG) < ∞ (iv) G satisfies b(m), for some non-negative integer m.

Proof. It is clear that (ii)⇒(iii)⇒(iv)⇒(i). If G is of type Φ then as B(G, ℤ) is a free ℤH-module for every finite subgroup H of G [6], it follows that proj. dim_{ℤG} B(G, ℤ) < ∞. Hence by Proposition 2.1 if G is of type Φ then G satisfies b(n_{0}).

Proposition 2.4.

(i) If G is of type Φ then so is every subgroup of G (ii) If G is a torsion-free group, then G is of type Φ iff cd_{ℤ}G < ∞.

Moreover, if cd_{ℤ}G < ∞ then cd_{ℤ}G = proj. dim_{ℤG}B(G, ℤ).

Proof. In Lemma 6.1 of [8] it is shown that b(n) is a subgroup closed property, so (i) follows. It is clear that if G is torsion-free then G is of type Φ iff cd_{ℤ}G < ∞. The rest follows form Proposition 2.1.

3. The Theorems and the Conjecture

Lemma 3.1. Let G be any group. Then

(i) If G is of type Φ then proj. dim_{ℤG}B(G, ℤ) < ∞ (ii) Let K be a subgroup of G, H a finite subgroup of G and W (H) = N_{G}(H)/H.

If proj. dim_{ℤG}B(G, ℤ) < ∞ then

proj. dim_{ℤK}B(K, ℤ) ⩽ proj. dim_{ℤG}B(G, ℤ) and proj. dim_{ℤW (H)}B(W (H), ℤ) ⩽ proj. dim_{ℤG}B(G, ℤ).

Proof. (i) follows from 2.3 (i)⇒(ii)

(ii) follows from Lemma 7.2 in [6].

Proof of Theorem 1.1. The proof follows essentially the proof of Theorem B in [6] by taking into account Lemma 3.1 and Proposition 2.1.

If G is a group such that N_{G}(H)/H is of type Φ for every finite subgroup H of G then it follows from Lemma 3.1 and Proposition 2.1 that G satisfies B(n_{0}), where n_{0} = proj. dim_{ℤG}B(G, ℤ). Hence from W. Lück’s Theorem 1.10 in [8] we obtain

Theorem 3.2. If G is a group such that dim | Λ(G) |< ∞ and N_{G}(H)/H is of type Φ for any finite subgroup H of G then G admits a model for \underline{E}G of dimension at most max{3, n_{0}} + λ(n_{0} + 1), where n_{0} = proj. dim_{ℤG}B(G, ℤ) and λ is a bound for the length l(H) of any finite subgroup H of G.

In [8] W. Lück showed that if G admits a finite dimensional model for \underline{E}G then G satisfies condition B(n), for some non-negative integer, which of course implies that N_{G}(H)/H is of type Φ for every finite subgroup H of G.

The finiteness of the dimension of | Λ(G) | is not a necessary condition for the group G to admit a finite dimensional model for \underline{E}G. If G is an infinite countable locally finite group, then G acts on a tree, without inversions, with finite stabilizers.

We propose the following as an algebraic characherization of those groups which admits a finite dimensional model for \underline{E}G.

Conjectrure A. The following statements are equivalent for a group G.

(1) G admits a finite dimensional model for \underline{E}G

(2) G is of type Φ

(3) spliℤG < ∞

(4) silpℤG < ∞

(5) fin. dim ℤG < ∞.

Proposition 3.3. The following implications hold in Conjecture A (1) ⇒ (2) ⇒ (3) ⇒ (4) ⇒ (5).

Proof. (1) ⇒ (2) If G admits a finite dimensional \underline{E}G then the cellular chain complex of \underline{E}G is of the form

0 ⟶ ⊕_{i_{σ}∈I_{σ}}ℤ (G/G_{i_{σ}}) ⟶ ⋯ ⟶ ⊕_{i_{0}∈I_{0}}ℤ (G/G_{i_{0}}) ⟶ ℤ ⟶ 0

where G_{i_{j}} are finite subgroups of G for all i_{j}.

If M is a ℤG-module which is of finite projective dimension as a ℤH-module for every finite subgroup H of G, then proj. dim_{ℤH}M ⩽ 1, hence

proj. dim_{ℤG}( ⊕_{i_{j}i∈I_{j}} ℤ (^{↘}G/G_{i_{j}})) ⊗ M^{↘} ⩽ 1

for all j = 0, … σ. Thus from the exact sequence of ℤG-modules

0 ⟶ ⊕_{i_{σ}∈I_{σ}}ℤ (^{↘}G/G_{i_{σ}}) ⊗ M^{↘} ⟶ ⋯ ⟶ ⊕_{i_{0}∈I_{0}}ℤ (^{↘}G/G_{i_{0}}) ⊗ M^{↘} ⟶ M ⟶ 0

follows that proj. dim_{ℤG}M ⩽ 1 + σ, so we obtain (2). In Proposition 2.2 it was shown that (2) ⇒ (3) ⇒ (4) ⇒ (5).

The next result is the first step to give us that 5) ⇒ 1) in Conjecture A, if G is a torsion-free locally soluble group.

Definition. Let S be the class of soluble groups G with the property, if K is a subgroup of G with Hirsch number, h(K), finite then there is a torsion-free subgroup K_{1} of K with h(K_{1}) = h(K).

Note that S contains the torsion-free soluble groups, the abelian groups and the polycyclic groups.

Theorem 3.4. If G is a group in S with fin. dimℤG < ∞, then h(G) < ∞.

For the proof of Theorem 3.4 we’ll need

Lemma 3.5. If G is a group with fin. dimℤG < ∞ then

(i) If K ⩽ G and cd_{ℤ}K < ∞ then cd_{ℤ}K ⩽ fin. dimℤG (ii) If K ⩽ G then fin. dim ℤK ⩽ fin. dim ℤG.

Proof. (i) Let fin. dimℤG = n and K ⩽ G with cd_{ℤ}K = m. If

0 ⟶ P_{m} ⟶ ⋯ ⟶ P_{1} ⟶ P_{0} ⟶ ℤ ⟶ 0

is a ℤK-projective resolution of the trivial ℤK-module ℤ of length m then

0 ⟶ ℤ^{↘}G ⊗_{ℤK} P_{m} ⟶ ⋯ ⟶ ℤ^{↘}G ⊗_{ℤK} P_{1} ⟶ ℤ^{↘}G ⊗_{ℤK} P_{0} ⟶ ℤ^{↘}G ⊗_{ℤK} ℤ ⟶ 0

is a ℤG-projective resolution of the ℤG-module ℤ^{↘}G ⊗_{ℤK} ℤ of length m, hence m ⩽ n.

(ii) The proof of (ii) is similar to the proof ot (i).

Proof of Theorem 3.4. We’ll prove the theorem by induction on the derived length of G.

If G is abelian then by 3.5 (i) the rank of any free abelian subgroup of G is bounded by fin. dimℤG, hence h(G) < ∞.

Now let

1 ⟶ K ⟶ G ⟶^{π} G_{1} ⟶ 1  (∗)

be a group extension with derived length of K less than the derived length of G and G_{1} abelian.

Since fin. dimℤG < ∞ we have that fin. dimℤK < ∞ and thus by the induction hypothesis, h(K) < ∞.

Let A_{λ} be a free abelian subgroup of G_{1} of rank λ and G_{2} = π^{−1}(A_{λ}). Then from the group extension 1 → K → G_{2} ⟶^{π} A_{λ} → 1 follows that h(G_{2}) = h(K)+λ. Now let K_{2} be a torsion-free subgroup of G_{2} with h(K_{2}) = h(G_{2}) = h(K)+λ. Then h(K_{2}) = hd_{ℤ}K_{2} ⩽ cd_{ℤ}K_{2} and we know from [1] that h(K_{2}) ⩽ cd_{ℤ}K_{2} ⩽ h(K_{2}) + 1. Now by 3.6 (i) cd_{ℤ}K_{2} ⩽ fin. dimℤG, which implies that h(K) + λ ⩽ fin. dimℤG, thus there is a bound on the ranks of the free abelian subgroups of G_{1}. Hence h(G_{1}) < ∞ and from (∗) follows that h(G) = h(K) + h(G_{1}) < ∞.

This theorem generalizes a theorem of N. Petrosyan [10] which says that if G is a torsion-free soluble group, with the property, that there is an integer m such that if K is a subgroup of G with cd_{ℤ}K < ∞ then cd_{ℤ}K ⩽ m, then h(G) < ∞.

Corollary 3.6. If G is a torsion-free locally soluble group with fin. dim ℤG < ∞, then G admits a finite dimensional model for \underline{E}G.

Proof. It follows from Lemma 3.5 (i) that if A is an abelian subgroup of G then h(A) ⩽ fin. dimℤG, hence by a result of Merzljakov [9] G is soluble. Thus by Theorem 3.4 h(G) < ∞ which implies that cd_{Z}G < ∞ so G admits a finite dimensional K(G, 1)-complex Y. The universal cover of Y is a finite dimensional model for \underline{E}G.

References

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Olympia Talelli otalelli@cc.uoa.gr

Department of Mathematics University of Athens Panepistemiopolis, 15784 Athens Greece