INFINITE GROUPS WITH FIXED POINT PROPERTIES

G. ARZHANTSEVA, M.R. BRIDSON, T. JANUSZKIEWICZ, I. J. LEARY, A. MINASYAN, AND J. ŚWIA̧TKOWSKI

Abstract. We construct finitely generated groups with strong fixed point properties. Let X_{ac} be the class of Hausdorff spaces of finite covering dimension which are mod-p acyclic for at least one prime p. We produce the first examples of infinite finitely generated groups Q with the property that for any action of Q on any X ∈ X_{ac}, there is a global fixed point. Moreover, Q may be chosen to be simple and to have Kazhdan’s property (T). We construct a finitely presented infinite group P that admits no non-trivial action by diffeomorphisms on any smooth manifold in X_{ac}. In building Q, we exhibit new families of hyperbolic groups: for each n ≥ 1 and each prime p, we construct a non-elementary hyperbolic group G_{n,p} which has a generating set of size n + 2, any proper subset of which generates a finite p-group.

1. Introduction

We present three templates for proving fixed point theorems; two are based on relative small cancellation theory and one is based on the Higman Embedding Theorem. Each template demands as input a sequence of groups with increasingly strong fixed point properties. By constructing such sequences we prove the following fixed point theorems.

For p a prime, one says that a space is mod-p acyclic if it has the same mod-p Čech cohomology as a point. Let X_{ac} be the class of all Hausdorff spaces X of finite covering dimension such that there is a prime p for which X is mod-p acyclic. Let M_{ac} denote the subclass of smooth manifolds in X_{ac}.

Note that the class X_{ac} contains all finite dimensional contractible spaces and all finite dimensional contractible CW-complexes.

Theorem 1.1. There is an infinite finitely generated group Q that cannot act without a global fixed point on any X ∈ X_{ac}. If X ∈ X_{ac} is mod-p acyclic, then so is the fixed

2000 Mathematics Subject Classification. Primary 20F65, 20F67 Secondary 57Sxx. The work of Arzhantseva and Minasyan and a visit to Geneva by Świa̧tkowski were supported by

Swiss National Science Foundation (SNSF) grant PP002-68627; visits of Bridson to Geneva were also supported by the SNSF. The work of Świa̧tkowski was supported by the Polish Ministry of Science, Higher Education (MNiSW) grant N201 012 32/0718, and a visit by him to Ohio was supported by the Ohio State University’s Mathematics Research Insitute. Bridson was supported by a Senior Fellowship from the EPSRC of the United Kingdom and a Royal Society Wolfson Research Merit Award. Januszkiewicz and Leary were partially supported by NSF grants DMS-0706259 and DMS-0505471, respectively. Several of the authors worked on this manuscript at MSRI, Berkeley, where research is supported in part by NSF grant DMS-0441170.

point set for any action of Q on X. For any countable group C, the group Q can be chosen to have either the additional properties (i), (ii) and (iii) or (i), (ii) and (iii)′ described below:

(i) Q is simple; (ii) Q has Kazhdan’s property (T); (iii) Q contains an isomorphic copy of C; (iii)′ Q is periodic.

Since a countable group can contain only countably many finitely generated subgroups, it follows from property (iii) that there are continuously many (i.e., 2^{ℵ_{0}}) nonisomorphic groups Q with the fixed point property described in Theorem 1.1.

Note that Kazhdan’s property (T) of a countable group is equivalent to the fact that every isometric action of the group on a Hilbert space has a global fixed point.

No non-trivial finite group has a fixed point property as strong as the one in Theorem 1.1. Any finite group not of prime power order acts without a global fixed point on some finite dimensional contractible simplicial complex. Smith theory tells us that the fixed point set for any action of a finite p-group on a finite dimensional mod-p acyclic space is itself mod-p acyclic, but it is easy to construct an action of a non-trivial finite p-group on a 2-dimensional mod-q acyclic space without a global fixed point if q is any prime other than p. Since the fixed point property of Theorem 1.1 passes to quotients, it follows that none of the groups Q can admit a non-trivial finite quotient. This further restricts the ways in which Q can act on acyclic spaces. For example, if X ∈ X_{ac} is a locally finite simplicial complex and Q is acting simplicially, then the action of Q on the successive star neighbourhoods st_{n+1} := st(st_{n}(x)) of a fixed point x ∈ X must be trivial, because st_{n} is Q-invariant and there is no non-trivial map from Q to the finite group Aut(st_{n}). Since X = ⋃_{n} st_{n}, we deduce:

Corollary 1.2. The groups Q from Theorem 1.1 admit no non-trivial simplicial action on any locally-finite simplicial complex X ∈ X_{ac}.

There is a similar result to the above for certain sorts of actions on manifolds. However, a stronger result concerning triviality of actions on manifolds can be obtained more directly:

Proposition 1.3. A simple group G that contains, for each n > 0 and each prime p, a copy of (ℤ_{p})^{n} admits no non-trivial action by diffeomorphisms on any X ∈ M_{ac}. The group Q in Theorem 1.1 may be chosen to have this property.

Finite p-groups have a global fixed point whenever they act on compact Hausdorff spaces that are mod-p acyclic, but the groups Q do not have this property. Indeed, if Q is infinite and has property (T) then it will be non-amenable, hence the natural action of Q on the space of finitely-additive probability measures on Q will not have a global fixed point, and this space is compact, contractible, and Hausdorff.

We know of no finitely presented group enjoying the fixed point property described in Theorem 1.1. However, using techniques quite different from those used to construct the groups Q, we shall prove the following:

Theorem 1.4. There exists a finitely presented infinite group P that has no non-trivial action by diffeomorphisms on any smooth manifold X ∈ M_{ac}.

Theorem 1.1 answers a question of P. H. Kropholler, who asked whether there exists a countably infinite group G for which every finite-dimensional contractible G-CW-complex has a global fixed point. This question is motivated by Kropholler’s study of the closure operator H for classes of groups, and by the class H𝔉 obtained by applying this operator to the class 𝔉 of all finite groups [11]. Briefly, if ℭ is a class of groups, then the class Hℭ is the smallest class of groups that contains ℭ and has the property that if the group G admits a finite-dimensional contractible G-CW-complex X with all stabilizers already in Hℭ, then G is itself in Hℭ. Kropholler showed that any torsionfree group in H𝔉 of type F P_{∞} has finite cohomological dimension. Since Thompson’s group F is torsion-free and of type F P_{∞} but has infinite cohomological dimension [6], it follows that F is not in H𝔉. Until now, the only way known to show that a group is not in H𝔉 has been to show that it contains Thompson’s group as a subgroup. If Q is any of the groups constructed in Theorem 1.1, then Q has the property that for any class ℭ of groups, Q is in the class Hℭ if and only if Q is already in the class ℭ. In particular, Q is not in the class H𝔉. Note that many of the groups constructed in Theorem 1.1 cannot contain Thompson’s group F as a subgroup, for example the periodic groups.

Our strategy for proving Theorems 1.1 and 1.4 is very general. First, we express our class of spaces as a countable union X = ∪_{n∈ℕ}X_{n}. For instance, if all spaces in X are finite-dimensional, then X_{n} may be taken to consist of all n-dimensional spaces in X. Secondly, we construct finitely generated groups G_{n} that have the required properties for actions on any X ∈ X_{n}. Finally, we apply the templates described below to produce the required groups.

Template FP: ruling out fixed-point-free actions. If there is a sequence of finitely generated non-elementary relatively hyperbolic groups G_{n} such that G_{n} cannot act without a fixed point on any X ∈ X_{n}, then there is an infinite finitely generated group that cannot act without a fixed-point on any X ∈ X.

Template NA_{fg}: ruling out non-trivial actions. If there is a sequence of non-trivial finitely generated groups G_{n} such that G_{n} cannot act non-trivially on any X ∈ X_{n}, then there is an infinite finitely generated group that cannot act non-trivially on any X ∈ X.

Template NA_{fp}: finitely presented groups that cannot act. Let (G_{n}; ξ_{n,j}) (n ∈ ℕ, j = 1, …, J) be a recursive system of non-trivial groups and monomorphisms ξ_{n,j} : G_{n} → G_{n+1}. Suppose that each G_{n+1} is generated by ⋃_{j} ξ_{n,j}(G_{n}) and that for every m ∈ ℕ there exists n ∈ ℕ such that G_{n} cannot act non-trivially on any X ∈ X_{m}. Then there exists an infinite finitely presented group that cannot act non-trivially on any X ∈ X.

Only the first and third templates are used in the construction of the groups P and Q. We include the second template with a view to further applications.

The engine that drives the first two templates is the existence of common quotients established in Theorem 1.5 below. The proof of this theorem, given in Section 2, is based on the following result of Arzhantseva, Minasyan and Osin [1], obtained using small cancellation theory over relatively hyperbolic groups: any two finitely generated non-elementary relatively hyperbolic groups G_{1}, G_{2} have a common non-elementary relatively hyperbolic quotient H.

Theorem 1.5. Let {G_{n}}_{n∈ℕ} be a countable collection of finitely generated non-elemen-tary relatively hyperbolic groups. Then there exists an infinite finitely generated group Q that is a quotient of G_{n} for every n ∈ ℕ.

Moreover, if C is an arbitrary countable group, then such a group Q can be made to satisfy the following conditions

(i) Q is a simple group; (ii) Q has Kazhdan’s property (T); (iii) Q contains an isomorphic copy of C. If the G_{n} are non-elementary word hyperbolic groups, then claim (iii) above can be

replaced by (iii)′ Q is periodic.

This result immediately implies the validity of templates FP and NA_{fg}. Indeed, if G_{n} are the hypothesized groups of template FP, the preceding theorem furnishes us with a group Q that, for each n ∈ ℕ, is a quotient of G_{n} and hence cannot act without a fixed point on any X ∈ X_{n}. Now let G_{n} be the hypothesized groups of template NA_{fg}. They are not assumed to be relatively hyperbolic. We consider groups A_{n} := G_{n} ∗ G_{n} ∗ G_{n}, which also cannot act non-trivially on any X ∈ X_{n}. The group A_{n} is non-elementary and relatively hyperbolic as a free product of three non-trivial groups. Therefore, Theorem 1.5 can be applied to the sequence of groups A_{n}, providing a group Q_{1} which, as a quotient of A_{n}, cannot act non-trivially on any X ∈ X_{n} for any n ∈ ℕ.

Following the above strategy to prove Theorem 1.1, we first represent X_{ac} as a countable union X_{ac} = ∪_{n,p}X_{n,p}, where, for each prime number p, the class X_{n,p} consists of all mod-p acyclic spaces of dimension n. Then, in Section 3, we construct the groups required by template FP, proving the following result.

Theorem 1.6. For each n ∈ ℕ and every prime p, there exists a non-elementary word hyperbolic group G_{n,p} such that any action of G_{n,p} by homeomorphisms on any space X ∈ X_{n,p} has the property that the global fixed point set is mod-p acyclic (and in particular non-empty).

The mod-p acyclicity of the fixed point for the action of G_{n,p} on the space X is a consequence of the following (n, p)-generation property: there is a generating set S of G_{n,p} of cardinality n + 2 such that any proper subset of S generates a finite p-subgroup.

For certain small values of the parameters examples of non-elementary word hyperbolic groups with the (n, p)-generation property were already known (e.g., when n = 1 and p = 2 they arise as reflection groups of the hyperbolic plane with a triangle as a fundamental domain). Our construction works for arbitrary n and p. For large n it provides the first examples of non-elementary word hyperbolic groups possessing the (n, p)-generation property.

We construct the groups G_{n,p} as fundamental groups of certain simplices of groups all of whose local groups are finite p-groups. We use ideas related to simplicial nonpositive curvature, developed by Januszkiewicz and Świa̧tkowski in [9], to show that these groups are non-elementary word hyperbolic. The required fixed point property is obtained using Smith theory and a homological version of Helly’s theorem.

Thus, to complete the proof of Theorem 1.1 and Corollary 1.2, it remains to prove Theorems 1.5 and 1.6. This will be done in Sections 2 and 3, respectively.

The validity of template NA_{fp} will be established in Section 4; it relies on the Higman Embedding Theorem. Also contained in Section 4 is Lemma 4.5, which establishes a triviality property for actions on manifolds. This is used both to provide input to the template NA_{fp} and to prove Proposition 1.3.

2. Relatively hyperbolic groups and their common quotients

Our purpose in this section is to provide the background we need concering relatively hyperbolic groups and their quotients. This will allow us to prove Propositions 2.6 and 2.8 below, which immediately imply the assertion of Theorem 1.5. We adopt the combinatorial approach to relative hyperbolicity that was developed by Osin in [18].

Assume that G is a group, {H_{λ}}_{λ∈Λ} is a fixed collection of proper subgroups of G (called peripheral subgroups), and A is a subset of G. The subset A is called a relative generating set of G with respect to {H_{λ}}_{λ∈Λ} if G is generated by A ∪ ⋃_{λ∈Λ} H_{λ}. In this case G is a quotient of the free product

F = (∗_{λ∈Λ}H_{λ}) ∗ F (A),

where F (A) is the free group with basis A. Let R be a subset of F such that the kernel of the natural epimorphism F ↠ G is the normal closure of R in the group F. In this case we will say that G has the relative presentation

(1)  ⟨A, {H_{λ}}_{λ∈Λ} ∥ R = 1, R ∈ R⟩.

If the sets A and R are finite, the relative presentation (1) is said to be finite.

Set H = ⨆_{λ∈Λ}(H_{λ} \ {1}). A finite relative presentation (1) is said to satisfy a linear relative isoperimetric inequality if there exists C > 0 such that for every word w in the alphabet A ∪ H (for convenience, we will further assume that A^{−1} = A) representing the identity in the group G, one has

w =^{F} ∏_{i=1}^{k} f_{i}^{−1}R_{i}^{±1}f_{i},

with equality in the group F, where R_{i} ∈ R, f_{i} ∈ F, for i = 1, …, k, and k ≤ C∥w∥, where ∥w∥ is the length of the word w.

Definition 2.1. [18] The group G is said to be relatively hyperbolic if there is a collection {H_{λ}}_{λ∈Λ} of proper peripheral subgroups of G such that G admits a finite relative presentation (1) satisfying a linear relative isoperimetric inequality.

This definition is independent of the choice of the finite generating set A and the finite set R in (1) (see [18]).

The definition immediately implies the following basic facts (see [18]):

Remark 2.2. (a) Let {H_{λ}}_{λ∈Λ} be an arbitrary family of groups. Then the free product G = ∗_{λ∈Λ}H_{λ} will be hyperbolic relative to {H_{λ}}_{λ∈Λ}.

(b) Any word hyperbolic group (in the sense of Gromov) is hyperbolic relative to the family {{1}}, where {1} denotes the trivial subgroup.

The following result is our main tool for constructing common quotients of countable families of relatively hyperbolic groups. Recall that a group G is said to be nonelementary if it does not contain a cyclic subgroup of finite index.

Theorem 2.3. [1, Thm. 1.4] Any two finitely generated non-elementary relatively hyperbolic groups G_{1}, G_{2} have a common non-elementary relatively hyperbolic quotient H.

Consider a sequence of groups (G_{n})_{n∈ℕ} such that G_{i} = G_{1}/K_{i}, i = 2, 3, …, for some K_{i} ⊲ G_{1} and K_{i} ≤ K_{i+1} for all i ∈ ℕ, i ≥ 2. The direct limit of the sequence (G_{n})_{n∈ℕ} is, by definition, the group G_{∞} = G_{1}/K_{∞} where K_{∞} = ⋃_{n=2}^{∞} K_{n}.

Remark 2.4. If G_{1} is finitely generated and G_{n} is infinite for every n ∈ ℕ, then G_{∞} is also infinite.

Indeed, suppose that G_{∞} is finite, i.e., |G_{1} : K_{∞}| < ∞. Then K_{∞} is finitely generated as a subgroup of G_{1}, hence there exists m ∈ ℕ such that K_{∞} = K_{m}, and G_{∞} = G_{m} is infinite; this is a contradiction.

Remark 2.5. Any infinite finitely generated group G contains a normal subgroup N that is maximal with respect to the property |G : N| = ∞.

Indeed, let N be the set of all normal subgroups of infinite index in G ordered by inclusion. Consider a chain (M_{i})_{i∈I} in N. Set M = ∪_{i∈I}M_{i}; then, evidently, M ⊲ G. Now, if M had finite index in G, then it would also be finitely generated. Hence, by the definition of a chain, there would exist j ∈ I such that M = M_{j}, which would contradict the assumption |G : M_{j}| = ∞. Therefore M ∈ N is an upper bound for the chain (M_{i})_{i∈I}. Consequently, one can apply Zorn’s Lemma to achieve the required maximal element of N.

Proposition 2.6. Let {G_{i}}_{i∈ℕ} be a countable collection of finitely generated non-ele-mentary relatively hyperbolic groups and let C be an arbitrary countable group. Then there exists a finitely generated group Q such that (i) Q is a quotient of G_{i} for every i ∈ ℕ; (ii) Q is a simple group; (iii) Q has Kazhdan’s property (T); (iv) Q contains an isomorphic copy of C.

Proof. First, embed C into an infinite finitely generated simple group S (see [13, Ch. IV, Thm. 3.5]). Let S′ be a copy of S. Then the group K = S ∗ S′ will be non-elementary and hyperbolic relative to the family consisting of two subgroups {S, S′}. Take G_{0} to be an infinite word hyperbolic group that has property (T). Then G_{0} is non-elementary and relatively hyperbolic by Remark 2.2, hence we can use Theorem 2.3 to find a nonelementary relatively hyperbolic group G(0) that is a common quotient of K and G_{0} (in particular, G(0) will also be finitely generated). Now, apply Theorem 2.3 to the groups G(0) and G_{1} to obtain their common non-elementary relatively hyperbolic quotient G(1). Similarly, define G(i) to be such a quotient for the groups G(i − 1) and G_{i}, i = 2, 3, . . .. Let G(∞) be the direct limit of the sequence (G(i))_{i=0}^{∞}.

The group G(∞) is finitely generated (as a quotient of G(0)) and infinite (by Remark 2.4), therefore, by Remark 2.5, there exists a normal subgroup N ⊲ G(∞) that is maximal with respect to the property |G(∞) : N| = ∞. Set Q = G(∞)/N. Then Q is an infinite group which has no non-trivial normal subgroups of infinite index. Being a quotient of G(0), makes Q a quotient of K = S ∗S′, therefore it can not have any proper subgroups of finite index. Thus, Q is simple. Since the homomorphism φ : S ∗ S′ → Q has a non-trivial image, it must be injective on either S or S′. Therefore Q will contain an isomorphically embedded copy of S, and, consequently, of C.

The property (i) for Q follows from the construction. The property (iii) holds because Q is a quotient of G_{0} and since Kazhdan’s property (T) is stable under passing to quotients. □

In the case when one has a collection of word hyperbolic groups (in the usual, nonrelative, sense), one can obtain common quotients with different properties by using Ol’shanskii’s theory of small cancellation over hyperbolic groups. For example, it is shown in [15] that if g is an element of infinite order in a non-elementary word hyperbolic group G, then there exists a number n > 0 such that the quotient of G by the normal closure of g^{n} is again a non-elementary word hyperbolic group. By harnessing this result to the procedure for constructing direct limits used in the proof of Proposition 2.6, we obtain the following statement, first proved by Osin:

Theorem 2.7. [19, Thm. 4.4] There exists an infinite periodic group O, generated by two elements, such that for every non-elementary word hyperbolic group H there is an epimorphism ρ : H ↠ O.

Proposition 2.8. There exists an infinite finitely generated group Q such that (a) Q is a quotient of every non-elementary word hyperbolic group;

(b) Q is a simple group;

(c) Q has Kazhdan’s property (T);

(d) Q is periodic.

Proof. Let O be the group given by Theorem 2.7. Since O is finitely generated, it has a normal subgroup N ⊲ O maximal with respect to the property |O : N| = ∞ (see Remark 2.5). Set Q = O/N. Then Q has no non-trivial normal subgroups of infinite index and is a quotient of every non-elementary word hyperbolic group; thus Q satisfies (a). In addition, Q is periodic since it is a quotient of O.

Observe that for an arbitrary integer k ≥ 2 there exists a non-elementary word hyperbolic group H = H(k) which does not contain any normal subgroups of index k (for instance, one can take H to be the free product of two sufficiently large finite simple groups, e.g., H = Alt(k + 3) ∗ Alt(k + 3)). Therefore the group Q, as a quotient of H, does not contain any normal subgroups of index k, for every k ≥ 2, hence it is simple. It satisfies Kazhdan’s property (T) because there are non-elementary word hyperbolic groups with (T) and property (T) is inherited by quotients. □

Remark 2.9. The method that we used to obtain simple quotients in the proofs of Propositions 2.6 and 2.8 was highly non-constructive as it relied on the existence of a maximal normal subgroup of infinite index provided by Zorn’s lemma. However, one can attain simplicity of the direct limit in a much more explicit manner, by imposing additional relations at each step. For word hyperbolic groups this was done in [14, Cor. 2]. The latter method for constructing direct limits of word hyperbolic groups was originally described by Ol’shanskii in [15]; it provides significant control over the resulting limit group. This control allows one to ensure that the group Q enjoys many properties in addition to the ones listed in the claim of Proposition 2.8. For example, in Proposition 2.8 one can add that Q has solvable word and conjugacy problems.

3. Simplices of finite p-groups with non-elementary word hyperbolic direct limits

Theorem 1.6 is an immediate consequence of the following two results, whose proof is the object of this section.

Theorem 3.1. For every prime number p and integer n ≥ 1 there is a non-elementary word hyperbolic group G generated by a set S of cardinality n + 2 such that the subgroup of G generated by each proper subset of S is a finite p-group.

Theorem 3.2. Let p be a prime number. Suppose that a group G has a generating set S of cardinality n + 2, such that the subgroup generated by each proper subset of S is a finite p-group. Then for any action of G on a Hausdorff mod-p acyclic space X of covering dimension less than or equal to n, the global fixed point set is mod-p acyclic.

We prove Theorem 3.1 by constructing each of the desired groups as the fundamental group (equivalently, the direct limit) of a certain (n + 1)-dimensional simplex of finite p-groups. The construction of the local groups in each simplex of groups, for fixed p, proceeds by induction on n. Provided that m ≤ n, the groups that are assigned to each codimension m face of the (n + 1)-simplex will depend only on m, up to isomorphism. The codimension zero face, i.e., the whole (n + 1)-simplex itself, will be assigned the trivial group 1, and each codimension one simplex will be assigned a cyclic group of order p. As part of the inductive step, we will show that the fundamental group of the constructed (n+1)-simplex of groups maps onto a p-group in such a way that each local group maps injectively. This quotient p-group will be the group used as each vertex group in the (n + 2)-simplex of groups.

The idea that drives our construction consists of requiring and exploiting existence of certain retraction homomorphisms between the local groups of the complexes of groups involved. We develop this approach in Subsections 3.2–3.4 below, after recalling in Subsection 3.1 some basic notions and facts related to complexes of groups.

Each simplex of groups that we construct will be developable. Associated to any developable n-simplex of groups G, there is a simplicial complex X on which the fundamental group G of G acts with an n-simplex as strict fundamental domain. If the local groups of G are all finite, the corresponding action is proper. Thus we may show that the group G is word hyperbolic by showing that the associated simplicial complex X is Gromov hyperbolic. We show that X is indeed hyperbolic by verifying that it satisfies a combinatorial criterion for the hyperbolicity of a simplicial complex related to the idea of simplicial non-positive curvature developed in [9]. More precisely, we show that X is 8-systolic, and hence hyperbolic. This is the content of Subsection 3.5.

¿From the perspective of the subject of simplicial non-positive curvature, Subsections 3.1-3.5 may be viewed as providing an alternative to the construction from [9] of numerous examples of k-systolic groups and spaces, for arbitrary k and in arbitrary dimension. The resulting groups are different from those obtained in [9].

¿From the algebraic perspective, this construction provides new operations of product type for groups, the so called n-retra-products, which interpolate between the direct product and the free product. These operations can be further generalized in the spirit of graph products. We think the groups obtained this way deserve further study. The groups obtained this way from finite groups fall in the class of systolic, or even 8systolic groups, and thus share various exotic properties of the latter, as established in [9, 10, 16, 17]. In a future work we plan to show that n-retra-products of finite groups, for sufficiently large n, are residually finite.

The last subsection of this section, Subsection 3.6, contains the proof of Theorem 3.2. This proof uses a result from Smith theory concerning mod-p cohomology of the fixed point set of a finite p-group action. It also uses a homological version of Helly’s theorem for mod-p acyclic subsets.

10 ARZHANTSEVA, BRIDSON, JANUSZKIEWICZ, LEARY, MINASYAN, AND ŚWIA̧TKOWSKI

Remark 3.3. It is because we need to apply Smith theory that the groups previously constructed in [9] are unsuitable for our purposes. The groups constructed in [9] include fundamental groups of simplices of finite groups which are non-elementary word hyperbolic, but the finite groups occurring in [9] are not of prime power order. One can show that the fundamental group of any (n + 1)-simplex of finite groups cannot act without a global fixed point by isometries on any complete CAT(0) space of covering dimension at most n. Indeed, the Helly-type argument goes through almost unchanged, while the fact that the fixed point set for a finite group of isometries of a complete CAT(0) space is contractible replaces the appeal to Smith theory. This argument originates in unpublished work of Farb, and was extended by Bridson (also unpublished). A special case appears in [2].

3.1. Strict complexes of groups. We recall some basic notions and facts related to strict complexes of groups. The main reference is Bridson and Haefliger [4], where these objects are called simple complexes of groups.

A simplicial complex K gives rise to two categories: the category Q_{K} of non-empty simplices of K with inclusions as morphisms, and the extended category Q_{K}^{+} of simplices of K including the empty set ∅ as the unique (−1)–simplex. In addition to the morphisms from Q_{K}, the category Q_{K}^{+} has one morphism from ∅ to σ for each nonempty simplex σ of K. A strict complex of groups G consists of a simplicial complex |G| (called the underlying complex of G), together with a contravariant functor G from Q_{|G|} to the category of groups and embeddings. A strict complex of groups is developable if the functor G extends to a contravariant functor G^{+} from the category Q_{|G|}^{+} to the category of groups and embeddings. Given an extension G^{+} of G, we will denote by G the group G^{+}(∅). For simplices τ ⊂ σ (allowing τ = ∅), we will view the group G(σ) as subgroup in the group G(τ). We will be interested only in extensions that are surjective, i.e. such that the group G = G^{+}(∅) is generated by the union of its subgroups G(σ) with σ ≠ ∅.

We call any surjective extension G^{+} of G an extended complex of groups. We view the collection of all possible surjective extensions of G to Q_{|G|}^{+} also as a category, which we denote by Ext_{G}. We take as morphisms of Ext_{G} the natural transformations from G^{+} to G^{+}′ which extend the identical natural transformation of G. (Note that, given extensions G^{+} and G^{+}′, there may be no morphism between them, and if there is one then, by surjectivity, it is unique; moreover, the homomorphism from G = G^{+}(∅) to G′ = G^{+}′(∅) induced by a morphism is not required to be an embedding, although it is required to be a group homomorphism.) If G is developable, the category Ext_{G} has an initial object G_{dir}^{+}, in which the group G_{dir}^{+}(∅) is just the direct limit of the functor G (for brevity, we often denote this direct limit group G̃). Thus for any extension G^{+} of G, there is a unique group homomorphism from G̃ to G extending the identity map on each G(σ) for σ ∈ Q_{|G|}. In the cases that will be considered below, the simplicial complex |G| is simply connected, which implies that G̃ coincides with what is known as the fundamental group of the complex of groups G. (In fact, |G| is contractible in the cases considered below.)

For an extended complex of groups G^{+} we consider a space dG^{+} with an action of G = G^{+}(∅), the development of G^{+}, given by

dG^{+} = |G| × G/_{∼},

where the equivalence relation ∼ is given by (p, g) ∼ (q, h) iff p = q and there exists σ ∈ Q_{|G|} so that p ∈ σ and g^{−1}h ∈ G(σ). It suffices to take σ to be the minimal simplex containing p. The G-action is given by g[p, h] = [p, gh]. The quotient by the action of G is (canonically isomorphic with) |G|, and the subcomplex

[|G|, 1] = {[(p, 1)] : p ∈ |G|},

(where [(p, 1)] is the equivalence class of (p, 1) under ∼) is a strict fundamental domain for the action (in the sense that every G-orbit intersects [|G|, 1] in exactly one point). The space dG^{+} is a multi-simplicial complex, and the (pointwise and setwise) stabilizer of the simplex [σ, g] is the subgroup gG(σ)g^{−1}. In the cases considered below, developments will be true simplicial complexes.

A morphism φ from a strict complex of groups G to a group H is a compatible collection of homomorphisms φ_{σ} : G(σ) → H, σ ∈ Q_{|G|} (in general not necessarily injective). Compatibility means that we have equalities φ_{σ} = φ_{τ} ∘ i_{στ} for any τ ⊂ σ, where i_{στ} is the inclusion of G(σ) in G(τ). For example, a collection of inclusions G(σ) → G^{+}(∅) is a morphism G → G^{+}(∅). A morphism φ : G → H is locally injective if all the homomorphisms φ_{σ} are injective.

Suppose we are given an action of a group H on a simplicial complex X, by simplicial automorphisms, and suppose this action is without inversions, i.e., if g ∈ H fixes a simplex of X, it also fixes all vertices of this simplex. Suppose also that the action has a strict fundamental domain D which is a subcomplex of X. Clearly, D is then isomorphic to the quotient complex H\X. Such an action determines the extended associated complex of groups G^{+}, with the underlying complex |G| = D, with local groups G(σ) := Stab(σ, H) for σ ⊂ D, and with G^{+}(∅) := H. The morphisms in G are the natural inclusions. It turns out that in this situatio the development dG^{+} is H-equivariantly isomorphic with X.

3.2. Higher retractibility. Now we pass to a less standard part of the exposition. We begin by describing a class of simplicial complexes, called blocks, that will serve as the underlying complexes of the complexes of groups involved in our construction. Then we discuss various requirements on the corresponding complexes of groups. Some part of this material is parallel to that in Sections 4 and 5 of [8], where retractibility and extra retractibility stand for what we call in this paper 1-retractibility and 2-retractibility, respectively.

Definition 3.4 (Block). A simplicial complex K of dimension n is a chamber complex if each of its simplices is a face of an n-simplex of K. Top dimensional simplices are then called chambers of K. A chamber complex K is gallery connected if each pair of its chambers is connected by a sequence of chambers in which any two consecutive chambers share a face of codimension 1. A chamber complex is normal if it is gallery connected and all of its links (which are also chamber complexes) are gallery connected. The boundary of a chamber complex K, denoted ∂K, is the subcomplex of K consisting of all those faces of codimension 1 that are contained in precisely one chamber. A block is a normal chamber complex with nonempty boundary. The sides of a block B are the faces of codimension 1 contained in ∂B. We denote the set of all sides of B by S_{B}.

Note that links B_{σ} of a block at faces σ ⊂ ∂B are also blocks, and that ∂(B_{σ}) = (∂B)_{σ}.

Definition 3.5 (Normal block of groups). A normal block of groups over a block B is a strict complex of groups G with |G| = B satisfying the following two conditions:

(1) G is boundary supported, i.e. G(σ) = 1 for each σ not contained in ∂B;

(2) G is locally S-surjective, i.e., every group G(σ) is generated by the union ⋃{G(s) : s ∈ S_{B}, σ ⊂ s}, where we use the convention that the empty set generates the trivial group 1.

An extended normal block of groups is an extension G^{+} of a normal block of groups G such that the associated morphism φ : G → G(∅) is S-surjective, i.e. G(∅) is generated by the union ⋃{G(s) : s ∈ S_{B}}. (To simplify notation, we write G(σ) instead of G^{+}(σ) to denote the corresponding groups of G^{+}.)

Given an extended normal block of groups G^{+} over B, its development dG^{+} is tesselated by copies of B. More precisely, dG^{+} is the union of the subcomplexes of the form [B, g], with g ∈ G(∅), which do not intersect each other except at their boundaries, and which we view as tiles of the tesselation. Moreover, the action of G(∅) on dG^{+} is simply transitive on these tiles. By S-surjectivity of φ, dG^{+} is a normal chamber complex. If |G(s)| > 1 for all s ∈ S_{B} then the chamber complex dG^{+} has empty boundary. If B is a pseudo-manifold and |G(s)| ≤ 2 for all s ∈ S_{B}, then dG^{+} is a pseudo-manifold.

Definition 3.6 (1-retractibility). An extended normal block of groups G^{+} is 1-retrac-tible if for every σ ⊂ |G| there is a homomorphism r_{σ} : G(∅) → G(σ) such that r_{σ}|_{G(s)} = id_{G(s)} for s ∈ S_{|G|}, s ⊃ σ, and r_{σ}|_{G(s)} = 1 otherwise.

In the next two lemmas we present properties that immediately follow from 1-retracti-bility. We omit the straightforward proofs.

Lemma 3.7.

(1) The homomorphisms r_{σ}, if they exist, are unique.

(2) Let φ_{σ} : G(σ) → G(∅) be the homomorphisms of the morphism φ : G → G(∅). Then for each σ we have r_{σ}φ_{σ} = id_{G(σ)}. Thus r_{σ} is a retraction onto the subgroup G(σ) < G(∅).

(3) The inclusion homomorphisms φ_{τσ} : G(τ) → G(σ), for σ ⊂ τ, occuring as the structure homomorphisms of G, satisfy φ_{τσ} = r_{σ}φ_{τ}.

Motivated by property (3) above, we define homomorphisms r_{ρτ} : G(ρ) → G(τ), for any simplices ρ, τ of |G|, including ∅, by putting r_{ρτ} := r_{τ}φ_{ρ}.

Lemma 3.8. Each of the homomorphisms r_{ρτ} is uniquely determined by the following two requirements:

(1) r_{ρτ}|_{G(s)} = id_{G(s)} for s ∈ S_{|G|}, s ⊃ ρ, s ⊃ τ;

(2) r_{ρτ}|_{G(s)} = 1 otherwise (i.e. for s ∈ S_{|G|}, s ⊃ ρ, s not containing τ). In particular, we have r_{σ∅} = φ_{σ}, r_{∅σ} = r_{σ}, and r_{τσ} = φ_{τσ} whenever τ ⊃ σ. Moreover, if τ ⊃ σ then r_{τσ} is a retraction (left inverse) for the inclusion φ_{τσ}.

To define higher retractibility properties for an extended normal block of groups G^{+} we need first to introduce certain new blocks of groups called unfoldings of G^{+} at the boundary simplices σ ⊂ ∂|G|.

Definition 3.9 (Unfolding of G^{+} at σ). Let G^{+} be a 1-retractible extended normal block of groups. Let σ ⊂ ∂|G| be a simplex, and denote by d_{σ}G(∅) the kernel of the retraction homomorphism r_{σ} : G(∅) → G(σ). The unfolding of G^{+} at σ, denoted d_{σ}G, is the complex of groups associated to the action of the group d_{σ}G(∅) on the development dG^{+}. The extended unfolding d_{σ}G^{+} is the same complex of groups equipped with the canonical morphism to the group d_{σ}G(∅).

Define a subcomplex d_{σ}|G| ⊂ dG^{+} by d_{σ}|G| := ⋃{[B, g] : g ∈ G(σ)}. We will show that the above defined unfolding d_{σ}G^{+} is an extended normal block of groups over d_{σ}|G|. This will be done in a series of lemmas, in which we describe the structure of d_{σ}|G| and d_{σ}G^{+} in detail.

Lemma 3.10. d_{σ}|G| is a strict fundamental domain for the action of the group d_{σ}G(∅) on the development dG^{+}. In particular, |d_{σ}G| = d_{σ}|G|.

Proof. We need to show that the restriction to d_{σ}|G| of the quotient map q_{σ} : dG^{+} → d_{σ}G(∅)\dG^{+} is a bijection. This follows by observing that the map j_{σ} : d_{σ}G(∅)\dG^{+} → d_{σ}|G| defined by j_{σ}(d_{σ}G(∅) · [p, g]) = [p, r_{σ}(g)] is the inverse of q_{σ}|_{d_{σ}|G|}. □

To proceed with describing d_{σ}|G|, we need to define links for blocks of groups. This notion will also be useful in our later considerations.

Definition 3.11 (Link of a block of groups). Let G be a normal block of groups and let σ be a simplex of |G|. The link of G^{+} at σ is an extended normal block of groups G_{σ}^{+} over the link |G|_{σ} given by G_{σ}^{+}(τ) := G(τ ∗σ) for all τ ⊂ |G|_{σ}, including the empty set ∅ (with the convention that ∅ ∗ σ = σ).

We skip the straightforward argument for showing that the above defined extended complex of groups is an extended normal block of groups.

The next lemma describes the links of the complex d_{σ}|G|. We omit its straightforward proof. In this lemma, and in the remaining part of this section, we will denote by σ − τ the face of σ spanned by the vertices of σ not contained in τ.

Lemma 3.12. Let [τ, g] be a simplex of d_{σ}|G|, where τ ⊂ |G| and g ∈ G(σ). For any simplex ρ ⊂ |G|_{τ} let d_{ρ}|G_{τ}| be the strict fundamental domain for the action of the group d_{ρ}G_{τ}(∅) on the development dG_{τ}^{+}. Then the link of d_{σ}|G| at [τ, g] has one of the following two forms depending on τ:

(1) (d_{σ}|G|)_{[τ,g]} ≅ d_{σ−τ}|G_{τ}| if σ and τ span a simplex of |G|, where we use convention that d_{∅}|G_{τ}| = dG_{τ}^{+}; (2) (d_{σ}|G|)_{[τ,g]} ≅ |G|_{τ} otherwise.

Lemma 3.12 easily implies the following corollary. The proof of part (1) uses induction on the dimension of B and S-surjectivity of the extending morphism; we omit the details.

Corollary 3.13.

(1) d_{σ}|G| is a normal chamber complex.

(2) The boundary ∂(d_{σ}|G|) is the subcomplex of d_{σ}|G| consisting of the simplices of form [ρ, g] for all ρ ⊂ ∂|G| not containing σ and for all g ∈ G(σ). In particular, the set of sides of d_{σ}|G| is the set

S_{d_{σ}|G|} = {[s, g] : s ∈ S_{|G|}, s does not contain σ, g ∈ G(σ)}.

For a subgroup H < G and an element g ∈ G, we denote by H^{g} the conjugation gHg^{−1}. The next lemma describes the local groups of the unfolding d_{σ}G.

Lemma 3.14. Let d_{σ}G be the unfolding of G and let [τ, g] be a simplex of d_{σ}|G| (with τ ⊂ |G| and g ∈ G_{σ}). Then

d_{σ}G([τ, g]) = [ker(r_{τσ} : G(τ) → G(σ))]^{g} < d_{σ}G(∅),

and consequently

d_{σ}G([τ, g]) = g · ⟨⋃ G(s) : s ∈ S_{|G|}, s ⊃ τ, s does not contain σ⟩ · g^{−1}.

In view of Corollary 3.13(2), Lemma 3.14 implies the following.

Corollary 3.15. For simplices [τ, g] of d_{σ}|G| not contained in the boundary ∂(d_{σ}|G|) we have d_{σ}G([τ, g]) = 1.

Another consequence of Lemma 3.14, which will be useful later, is the following.

Corollary 3.16. Let G^{+} be an extended normal block of groups, and let σ be a simplex of |G|. Then for any simplex [τ, g] ⊂ d_{σ}|G|, with τ ⊂ |G| and g ∈ G(σ), we have [d_{σ}G]_{[τ,g]} ≅ d_{σ−τ}G_{τ}^{+}, where ≅ denotes an isomorphism of extended complexes of groups, and where d_{∅}G_{τ}^{+} denotes here the trivial strict complex of groups over dG_{τ}^{+} (i.e. all of the local groups are trivial).

As a consequence of the results above, from Lemma 3.10 to Corollary 3.15, we obtain the following.

Corollary 3.17. Given an extended normal block of groups G^{+}, each of its unfoldings d_{σ}G^{+} is an extended normal block of groups.

We are now in a position to define recursively higher retractibilities.

Definition 3.18 (n-retractibility). Let n be a natural number. An extended normal block of groups G^{+} is (n + 1)-retractible if it is 1-retractible, and for every simplex σ ⊂ ∂|G| the unfolding d_{σ}G^{+} is n-retractible.

Example 3.19 (n-retractible 1-simplex of groups). Consider the extended complex of groups G^{+} with |G| equal to a 1-simplex, with the vertex groups G(v) cyclic of order two, and with G(∅) dihedral of order 2k, where generators of the vertex groups correspond to standard generators of the dihedral group. This complex of groups is clearly an extended normal block of groups. Moreover, it is n-retractible but not (n+1)-retractible in the case when k = 2^{n}(2m + 1) for some m.

Remark 3.20.

(1) Note that an n-retractible extended normal block of groups is always k-retrac-tible for each k ≤ n.

(2) All links in a 1-retractible normal block of groups are 1-retractible.

(3) From Corollary 3.16 and the above remark (2) one can easily deduce using induction on n that if G^{+} is n-retractible and σ is a simplex of |G|, then G_{σ}^{+} is also n-retractible.

3.3. n-retractible extensions. Our next objective is to establish results that give partial converses to property in Remark 3.20(3). These will allow us to pass up one dimension in our recursive construction of n-retractible simplices of groups, in the next subsection.

Proposition 3.21. Let G be a normal block of groups. If for some natural number n all the links in G (as extended complexes of groups) are n-retractible then:

(1) G is developable, and

(2) the extension G_{dir}^{+} of G (in which G(∅) coincides with the direct limit G̃ of G) is n-retractible.

Proof. To prove part (1), we need to show that G admits a locally injective morphism ψ : G → H to some group H. To find ψ, for each σ ⊂ |G| we construct a morphism r̄_{σ} : G → G(σ) which is identical on the group G(σ) of G. We then take as H the direct product H = ⊕{G(σ) : σ ⊂ |G|} and as ψ the direct product morphism ψ = ⊕{r̄_{σ} : σ ⊂ |G|}, which is then clearly locally injective.

Fix a simplex σ ∈ |G|. To get a morphism r̄_{σ} as above, we will construct an appropriate compatible collection of homomorphisms r̄_{ησ} : G(η) → G(σ), for all η ⊂ |G|, such that r̄_{σσ} = id_{G(σ)}. To do this, for any simplex ρ ⊂ |G| consider the set of sides S_{ρ} = {s ∈ S_{|G|} : ρ ⊂ s}. Fix a simplex η ⊂ |G|. If S_{η} ∩ S_{σ} = ∅, put r̄_{ησ} to be the trivial homomorphism. Otherwise, consider the simplex τ = ∩{s : s ∈ S_{η} ∩ S_{σ} }. Clearly, we have then η ⊂ τ and σ ⊂ τ. In particular, we have the inclusion homomorphism i_{τσ} : G(τ) → G(σ), which is identical on the subgroups G(s) : s ∈ S_{η} ∩ S_{σ}.

Recall that we denote by τ − η the face of τ spanned by all vertices not contained in η. Since η ⊂ τ, the groups G(η) and G(τ) coincide with the link groups G_{η}^{+}(∅) and G_{η}^{+}(τ − η), respectively. Since the link G_{η}^{+} is 1-retractible, we have the retraction

r_{τ−η} : G_{η}^{+}(∅) → G_{η}^{+}(τ − η)

such that r_{τ−η}|_{G_{η}^{+}(s)} = id_{G_{η}^{+}(s)} for s ∈ S_{|G|_{η}}, s ⊃ τ − η and r_{τ−η}|_{G_{η}^{+}(s)} = 1 otherwise (cf. Definition 3.6).

Put r̄_{ησ} := i_{τσ} ∘ r_{τ−η}. We claim that r̄_{ησ} satisfies the assertions (1) and (2) of Lemma 3.8, when substituted for r_{ησ}. This follows from the identification of the groups G(s), s ∈ S_{|G|} with the groups G_{η}^{+}(s−η), for η ⊂ s, and from the fact that s−η ∈ S_{|G|_{η}} (because ∂(|G|_{η}) = (∂|G|)_{η}).

Now, we need to check the compatibility condition r̄_{η_{2}σ} = r̄_{η_{1}σ} ∘ i_{η_{2}η_{1}} for all simplices η_{1} ⊂ η_{2} in |G|. This follows from the coincidence of the maps on both sides of the equality on the generating set ∪{G(s) : s ∈ S_{|G|}, s ⊃ η_{2}, } of G(η_{2}). This coincidence is a fairly direct consequence of assertions (1) and (2) of Lemma 3.8, satisfied by the maps r̄_{η_{1}σ} and r̄_{η_{2}σ}.

Finally, assertion (1) of Lemma 3.8 clearly implies that r̄_{σσ} = id_{G(σ)}, which concludes the poof of developability of G.

We now turn to proving part (2). To deal with the case n = 1 we need to construct the map r_{σ} : G̃ → G(σ) as required in Definition 3.6, for any σ ⊂ |G|.

Consider the maps r̄_{ησ} : η ⊂ |G| constructed in the proof of part (1), and the morphism r̄_{σ} : G → G(σ) given by these maps. Let r_{σ} : G̃ → G(σ) be the homomorphism induced by this morphism. The reqirements of Definition 3.6 for r_{σ} follow then easily from the assertions of Lemma 3.8 satisfied by the maps r̄_{ησ} (we skip the straightforward details). Thus 1-retractibility of links of G implies 1-retractibility of G_{dir}^{+}.

Now suppose that n > 1. If links in G are n-retractible, it follows that links in d_{σ}G are (n − 1)-retractible for all σ. By induction, it follows that the unfoldings d_{σ}G_{dir}^{+} are (n − 1)-retractible, and so G_{dir}^{+} is n-retractible. □

Example 3.22 (n-retractible 2-simplex of groups). Consider the triangle Coxeter group W_{n,m} given by

W_{n,m} = ⟨s_{1}, s_{2}, s_{3}|s_{i}^{2}, (s_{i}s_{j})^{k} for i ≠ j⟩,

where k = 2^{n}(2m + 1). The Coxeter complex for this group is a triangulation of the (hyperbolic) plane on which the group acts by simplicial automorphisms, simply transitively on 2-simplices. Let G^{+} be the extended 2-simplex of groups associated to this action. Links in G at vertices are then isomorphic to the 1-simplex of groups from Example 3.18. Thus, in view of Proposition 3.21 and Remark 3.20(3), G^{+} is n-retractible, but not (n + 1)-retractible.

A more subtle way of getting n-retractible extensions is given in the following theorem, which will be used directly, as a recursive step, in our construction in Subsection 3.4.

Theorem 3.23. Let G be a normal block of groups in which all links are n-retractible. Then there exists an extension G_{min}^{+} of G that has the following properties.

(1) G_{min}^{+} is the minimal n-retractible extension of G in the following sense: if G^{+} is any n-retractible extension of G then there is a unique morphism of extended complexes of groups G^{+} → G_{min}^{+} which extends the identity on G (i.e. a morphism in the category Ext_{G}).

(2) If all G(σ) are finite, so is G_{min}(∅).

(3) If all G(σ) are p-groups of bounded exponent, so is G_{min}(∅).

(4) If all G(σ) are soluble groups with soluble length ≤ d for some fixed d, then so is G_{min}(∅).

Remark 3.24. Property 1 means that the extended complex of groups G_{min}^{+} is the terminal object in the category of n-retractible extensions of G. In particular, it is unique.

Proof. Let G_{dir}^{+} be the direct limit extension of G, as described in Proposition 3.21. We recursively define iterated unfoldings of G_{dir}^{+}. For each simplex σ_{1} of |G|, define d_{σ_{1}}G_{dir}^{+} as previously. Suppose that a complex of groups d_{σ_{1},...,σ_{k}}G_{dir}^{+} has already been defined. For each simplex σ_{k+1} of the underlying simplicial complex |d_{σ_{1},...,σ_{k}}G_{dir}^{+}|, let d_{σ_{1},...,σ_{k},σ_{k+1}}G_{dir}^{+} = d_{σ_{k+1}}(d_{σ_{1},...,σ_{k}}G_{dir}^{+}). Since G is n-retractible, this allows us to define extended complexes of groups d_{σ_{1},...,σ_{k}}G_{dir}^{+} for any k ≤ n. Due to Corollary 3.17, all of these are extended normal blocks of groups.

Note that each of the groups d_{σ_{1},...,σ_{k}}G_{dir}(∅), which we denote G̃_{σ_{1},...,σ_{k}}, is a subgroup of the direct limit G̃ of G. Let N_{k} be the largest normal subgroup of G̃ which is contained in every subgroup G̃_{σ_{1},...,σ_{k}}, where σ_{1}, …, σ_{k} ranges over all allowed sequences of simplices (i.e., all sequences for which d_{σ_{1},...,σ_{k}}G_{dir}^{+} was defined above). If we set G_{min}(∅) = G̃/N_{n}, the resulting extension is clearly n-retractible. We need to show it is minimal.

Let G_{0}^{+} be any n-retractible extension of G. Clearly, we have the canonical surjective homomorphism h : G̃ → G_{0}(∅), which allows us to express G_{0}(∅) as the quotient G̃/ ker h. To get the homomorphism G_{0}(∅) → G_{min}(∅) = G̃/N_{n} as required for minimality, we need to show that ker h < N_{n}. By definition of N_{n}, it is thus sufficient to show that ker h < G̃_{σ_{1},...,σ_{n}} for all allowed sequences σ_{1}, …, σ_{n}.

To prove the latter, we will show by induction on k that ker h < G̃_{σ_{1},...,σ_{k}}, for all 1 ≤ k ≤ n. For k = 1, we have retractions G̃ → G(σ_{1}) and G_{0}(∅) → G(σ_{1}) which commute with h, and such that G̃_{σ_{1}} = ker[G̃ → G(σ_{1})] and d_{σ_{1}}G_{0}(∅) = ker[G_{0}(∅) → G(σ_{1})]. Consequently, we get a canonical homomorphism h_{σ_{1}} : G̃_{σ_{1}} → d_{σ_{1}}G_{0}(∅) with ker H_{σ_{1}} = ker h. In particular, ker h < G̃_{σ_{1}}.

To proceed, observe that (due to Lemma 3.14 applied recursively) the non-extended unfoldings d_{σ_{1},...,σ_{k}}G_{dir} and d_{σ_{1},...,σ_{k}}G_{0} coincide for all 1 ≤ k ≤ n. Thus, for any σ_{2} ⊂ |d_{σ_{1}}G_{dir}| = |d_{σ_{1}}G_{0}| we can repeat the above argument and get the homomorphism h_{σ_{1},σ_{2}} : G̃_{σ_{1},σ_{2}} → d_{σ_{1}σ_{2}}G_{0}(∅) with ker h_{σ_{1},σ_{2}} = ker h_{σ_{1}} = ker h. Repeating this argument, we finally get the homomorphisms h_{σ_{1},...,σ_{n}} : G̃_{σ_{1},...,σ_{n}} → d_{σ_{1},...,σ_{n}}G_{0}(∅) with ker h_{σ_{1},...,σ_{n}} = ker h. It follows that ker h < G̃_{σ_{1},...,σ_{n}}. This proves statement 1.

To prove statement 2, recall that, by Definition 3.4, the block |G| is finite. If G(σ) is finite for each σ ≠ ∅, then it follows (by applying recursively Lemmas 3.10 and 3.14) that each underlying simplicial complex |d_{σ_{1},...,σ_{k}}G_{dir}| is finite, and that for every nonempty simplex η ⊂ |d_{σ_{1},...,σ_{k}}G_{dir}| the group d_{σ_{1},...,σ_{k}}G_{dir}(η) is finite. Hence each G̃_{σ_{1},...,σ_{k}} has finite index in G̃, and there are finitely many such groups for each k ≤ n. If follows that the intersection of all such subgroups is a subgroup of finite index, and so each N_{k} for k ≤ n is a finite index normal subgroup of G̃. This proves that G_{min} = G̃/N_{n} is a finite group as claimed in statement 2.

Before proving statements 3 and 4, we first claim that each N_{k} is equal to the intersection of the groups G̃_{σ_{1},...,σ_{k}}. To see this, it is useful to change the indexing set. For τ_{1}, …, τ_{k} a sequence of simplices of dG_{dir}^{+} of length at most n, define σ_{1} to be the image of τ_{1} in |G| = G̃\dG_{dir}^{+}. Assuming that σ_{1}, …, σ_{i−1} have already been defined for some i with 1 < i ≤ k, define σ_{i} to be the image of τ_{i} in |d_{σ_{1},...,σ_{i−1}}G_{dir}^{+}| = G̃_{σ_{1},...,σ_{i−1}}\dG_{dir}^{+}. Also define d_{τ_{1}...,τ_{k}}G_{dir}^{+} to be equal to d_{σ_{1},...,σ_{k}}G_{dir}^{+}, and define G̃_{τ_{1}...,τ_{k}} to be equal to G̃_{σ_{1},...,σ_{k}}.

If x is a point of dG_{dir}^{+} whose stabilizer is some subgroup H < G̃, and if g is an element of G̃, the stabilizer of the point g.x is equal to the conjugation gHg^{−1}. This observation and induction show that for each g ∈ G̃, for each k ≤ n and for each sequence τ_{1}, …, τ_{k} of simplices of dG_{dir}^{+}, we have

G̃_{gτ_{1},...,gτ_{k}} = g · G̃_{τ_{1},...,τ_{k}} · g^{−1}.

Hence the intersection of the subgroups of the form G̃_{τ_{1},...,τ_{k}}, for fixed k ≤ n, is a normal subgroup of G̃. It follows that this intersection is equal to N_{k}.

The above observation combined with the inclusions G̃_{σ_{1},...,σ_{k}} < G̃_{σ_{1},...,σ_{k},σ_{k+1}} implies that N_{k+1} < N_{k}. To prove statement 3, we will show by induction that for 0 ≤ k < n the quotient groups N_{k}/N_{k+1} and G̃/N_{k+1} are p-groups of bounded exponent. Here we use convention that N_{0} = G̃.

For k = 0, we need to show that G̃/N_{1} is a p-group of bounded exponent. We know that N_{1} is the intersection of the groups of form G̃_{σ_{1}}, which are the kernels of retractions G̃ → G(σ_{1}). Thus G̃/N_{1} embeds in the product of the groups G(σ_{1}). Since the latter groups are p-groups of bounded exponent, the assertion follows.

Now we suppose G̃/N_{k} is a p-group of bounded exponent and claim that the group N_{k}/N_{k+1} is too. To see that this is true, recall that the groups G̃_{σ_{1},...,σ_{k+1}} are the kernels of the retractions G̃_{σ_{1},...,σ_{k}} → d_{σ_{1}...σ_{k}}G_{dir}(σ_{k+1}). Note also that

N_{k+1} = ⋂ G̃_{σ_{1},...,σ_{k+1}} = ⋂[N_{k} ∩ G̃_{σ_{1},...,σ_{k+1}}],

and thus N_{k+1} is equal to the intersection of the kernels of the composed homomorphisms

N_{k} ⟶^{incl} G̃_{σ_{1},...,σ_{k}} ⟶^{r} d_{σ_{1},...,σ_{k}}G_{dir}(σ_{k+1}).

Now, each of the groups d_{σ_{1},...,σ_{k}}G_{dir}(σ_{k+1}) canonically embeds in the quotient G̃/N_{k} (because the latter is a k-retractible extension of G̃). Thus all these groups are p-groups of finite exponent. As before, we see that N_{k}/N_{k+1} embeds in the product of p-groups of bounded exponents, hence the assertion. The fact that the quotient G̃/N_{k+1} is then also a p-group of bounded exponent follows directly. This proves statement 3.

The proof of statement 4 is similar to that of statement 3. □

3.4. The construction and retra-products. Given any n, we construct an n-retrac-tible extended simplex of groups G^{+}, over the simplex Δ of arbitrary dimension, as follows:

1. We put the trivial group on the simplex Δ.

2. We put an arbitrary group G(s) on each codimension 1 face s of Δ.

3. Suppose we have already defined groups G(η) (and inclusion maps between them) for faces of codimension strictly less than k. Then for a face τ of codimension k ≤ dim Δ, the group G(τ) is the minimal n-retractible extension, as in Theorem 20, of the simplex of groups over the link simplex Δ_{τ} made of the already defined groups G(η), via the canonical correspondence between the faces of Δ_{τ} and the faces η ⊂ Δ containing τ.

4. Finally, we take as G(∅) the minimal n-retractible extension of the so far obtained simplex of groups G over Δ.

Definition 3.25 (Retra-product). We will call the group G(∅) of any simplex of groups G^{+} obtained as in the construction above the n-retra-product of the (finite) family of groups G(s) : s ∈ S_{Δ}. Note that this operation makes sense for any finite family of groups.

Clearly, the groups G(τ) obtained in the construction above are all the n-retra-products of the corresponding families of groups G(s) : s ∈ S_{Δ}, s ⊃ τ.

Rephrasing Theorem 3.23 in the context of retra-products we get the following properties of this operation, for an arbitrary natural number n:

(1) the n-retra-product of finite groups is finite;

(2) the n-retra-product of p-groups of bounded exponent is a p-group of bounded exponent.

It follows that the n-retra-product of finite p-groups is a finite p-group. In particular, we get the following.

Corollary 3.26. Let G be a non-extended simplex of groups obtained as in the construction above, out of groups G(s) being finite p-groups. Then all groups G(τ) in this simplex are finite p-groups.

Remark 3.27. The construction of this subsection was first used in [8], in the 2-retractible case. For the purposes of this paper, it suffices to consider the case when each codimension one face of the simplex is assigned the cyclic group ℤ_{p} of order p. Even though the construction of the n-retra-products is in principle explicit, one rapidly loses track of the groups arising.

The n-retra-product of two groups ℤ_{2}, occuring at faces of codimension 2, is the dihedral group D_{2^{n}} of order 2^{n+1}. We do not even know the orders of the n-retra-products of three copies of ℤ_{2}, except the case n = 2 when this order is 2^{14}.

Note that if G is a non-extended simplex of groups corresponding to G^{+}, obtained by the construction above, then its direct limit extension G_{dir}^{+} is different from G^{+}, and in particular the direct limit G_{dir}^{+}(∅) is different from the n-retra-product G(∅). This motivates the following.

Definition 3.28 (Free retra-product). The direct limit of a non-extended simplex of groups G obtained by the construction above will be called the free n-retra-product of the (finite) family of the codimension 1 groups G(s) : s ∈ S_{Δ}.

In the next subsection we will deal with free n-retra-products of finite groups, showing that for n ≥ 2 they are infinite, and for n ≥ 3 they are non-elementary word-hyperbolic.

3.5. Simplicial non-positive curvature, word-hyperbolicity, and the proof of Theorem 3.1. To show that there are non-elementary word-hyperbolic groups arising from the construction of the previous subsection, we will use results of [9] concerning simplicial non-positive curvature.

Recall from [9] that the systole sys(K) of a simplicial complex K is the smallest number of 1-simplices in any full subcomplex of K homeomorphic to the circle. A simplicial complex K is k-large if its systole and the systoles of links of all simplices in K are all at least k. Simplicial complexes whose all links are k-large, for some fixed k, are the analogs of metric spaces with curvature bounded above. If they are additionally simply connected, we call them k-systolic complexes. All results of this subsection are corollaries to the following.

Proposition 3.29. Suppose G^{+} is an n-retractible extended simplex of groups. Then the development dG^{+} is 2(n + 1)-large.

We skip the proof of the proposition until the end of the subsection, first deriving (and making comments on) its consequences. In particular, we show how this proposition, together with the results of the previous subsection, implies Theorem 3.1.

Corollary 3.30. Suppose G is a non-extended simplex of groups whose links G_{σ}^{+} are n-retractible, and let G_{dir}^{+} be the direct limit extension of G.

(1) If n ≥ 2 then the development dG_{dir}^{+} is contractible.

(2) If n ≥ 2 and the codimension 1 groups G(s) are non-trivial then dG_{dir}^{+} and the direct limit G_{dir}(∅) are both infinite.

(3) If n ≥ 3 then the 1-skeleton of the development dG_{dir}^{+}, equipped with the polygonal metric, is Gromov-hyperbolic.

(4) If n ≥ 3 and all groups G(σ) are finite and nontrivial then the group G_{dir}(∅) is non-elementary word-hyperbolic, except when the underlying simplex is the 1-simplex and the vertex groups are of order 2 (in which case it is the infinite dihedral group).

Proof. First, note that links in dG_{dir}^{+} are isomorphic to the developments of the nretractible link simplices of groups G_{σ} (see Lemma 9(1)), and thus are 2(n + 1)-large.

Since dG_{dir}^{+} is the universal cover of G, it is simply connected and hence 2(n + 1)- systolic. For n ≥ 2, this means that dG_{dir}^{+} is 6-systolic, and thus it is contractible by [9, Theorem 4.1(1)]. This proves (1).

To get assertion (2), note that it follows from [9, Proposition 18(2)], that the group G_{dir}(∅) has a nontrivial subgroup that acts on the development dG_{dir}^{+} freely. Thus this development is a classifying space for this subgroup and, since it has finite dimension, the subgroup has to be infinite. See also [16, Corollary 4.3], for a more elementary argument.

For parts (3) and (4), note first that n ≥ 3 implies 8-systolicity of the development dG_{dir}^{+}. Part (3) follows then from [9, Theorem 2.1].

Finally, under the assumptions of (4), the group G_{dir}(∅) acts on dG_{dir}^{+} properly discontinuously and cocompactly. Thus it follows from (3) that G_{dir}(∅) is word-hyperbolic. If the underlying simplex of |G| is 1-dimensional, the group acts geometrically on the tree dG_{dir}^{+}, and hence is virtually free nonabelian (except the mentioned case). If the dimension of the underlying simplex |G| is greater than 1, the fact that the group is non-elementary follows from [17, Theorem 5.6 and Remark 2 at the end], where it is shown that this group has one end.

The non-elementarity above can be also shown directly, by noting that the groups as in (4) contain as a subgroup the free product of three nontrivial finite groups (namely codimension 1 groups in any vertex unfolding of certain three pairwise disjoint sides). We do not include the details of this argument. □

By specializing to free n-retra-products, we immediately get the following.

Corollary 3.31. If n ≥ 2 then the free n-retra-product of (at least two) nontrivial groups is infinite. If n ≥ 3 then the free n-retra-product of (at least two) nontrivial finite groups is non-elementary word-hyperbolic, except the product of two groups of order 2.

Remark 3.32. Note that the construction in the present paper gives new families of ksystolic groups, for arbitrary k, different from those constructed in [9, Sections 17–20]. These are the free n-retra-products of arbitrary finite groups, where 2(n + 1) ≥ k.

Proof of Theorem 3.1. Let G be the 3-retractible (n + 1)-simplex of groups obtained by the construction of Subsection 3.4, with all codimension 1 groups G(s) isomorphic to the cyclic group ℤ_{p} of order p. Let G be the direct limit of G, i.e. the free 3retra-product of n + 2 copies of ℤ_{p}. By Corollary 3.30, G is then non-elementary word hyperbolic.

Choose a generator for each codimension 1 subgroup G(s) of G (i.e. for each factor of the above free 3-retra-product). Let S be the set formed of these generators. Then S consists of n + 2 elements, and it generates G since the union ∪{G(s) : s ∈ |G|} generates G. For any proper subset T ⊂ S, the subgroup ⟨T ⟩ < G generated by T coincides with one of the local groups of G. More precisely, for τ = ∩{s : s ∈ T } we have ⟨T ⟩ = G(τ). By Corollary 3.25, ⟨T ⟩ is then a finite p-group, which completes the proof. □

It only remains now to prove Proposition 3.29. We will use the following lemma in the proof.

Lemma 3.33. Let γ be a loop in the 1-skeleton of a simplicial complex K which has the minimum length L amongst all loops in the 1-skeleton of K in the same free homotopy class. Then any lift γ̃ of γ to the universal cover K̃ of K has the property that it minimizes distance measured in the 1-skeleton of K̃ between any two points whose distance in γ̃ is at most L.

Proof. Let t be a deck transformation of K̃ that acts as a translation by the distance L on the subcomplex γ̃. Suppose that γ̃ does not have the property claimed. Then there exist vertices p and q of γ̃ such that q lies on the segment from p to tp and such that the distance between p and q in the 1-skeleton of K̃ is strictly smaller than the distance between them in γ̃. Let α be the segment of γ̃ between p and q, and let α′ be a path of shorter length in the 1-skeleton of K̃ from p to q. Denote also by β be the segment of γ̃ between q and tp. Since K̃ is simply connected, α and α′ are homotopic relative to their endpoints. This homotopy, after projecting to K, yields a homotopy between γ and the loop obtained by projecting α′ ∪ β. Since the latter loop is strictly shorter, we get a contradiction. □

Proof of Proposition 3.29. The proof is by induction on d, the dimension of the simplex |G|, followed by induction on n. For d = 0 there is nothing to prove. Let G^{+} be an n-retractible d-dimensional simplex of groups. Then links G_{σ}^{+} are also n-retractible (see Remark 17(3)), and thus by induction on d their developments dG_{σ}^{+} are 2(n + 1)-large. Thus the same holds for links of dG^{+}.

By [9, Corollary 1.5], a simplicial complex X with k-large links is k-large iff the length of the shortest loop in the 1-skeleton of X which is homotopically nontrivial in X is at least 2(n + 1). We thus need to show this for X = dG^{+}.

Let γ be a homotopically nontrivial polygonal loop in dG^{+} of the shortest length. Now we start the induction on n. It has been shown in [8, Proposition 4.3(3)], that the development of any 1-retractible extended simplex of groups is a flag complex. Hence the length of γ is at least 4. This completes the case n = 1 for all d. Clearly an n-retractible complex is (n − 1)-retractible, and so by induction on n, the length of γ is at least 2n.

Let X̃ be the universal cover of X = dG^{+}, let γ̃ be a lift of γ to X̃, and let v_{0}, …, v_{n+1} be some consecutive vertices on γ̃. By Lemma 3.33, we see that γ̃ minimizes distances between the vertices v_{i} : 0 ≤ i ≤ n + 1 in the 1-skeleton of X̃. In particular the distance between v_{0} and v_{n+1} is equal to n + 1. Let δ denote the segment of γ̃ between v_{0} and v_{n+1}, and let δ′ be the segment of γ̃ that starts at v_{n+1} and projects to the segment of γ complementary to the projection of δ.

By symmetry of dG^{+}, we may assume that v_{1} is a vertex of the simplex [|G|, 1] ⊂ dG^{+}. Consider the fundamental domain D ⊆ X̃ for the group G̃_{v_{1},...,v_{n}} = d_{v_{1},...,v_{n}}(∅) obtained recursively as d_{v_{n}} … d_{v_{1}}|G| in the way described just before Lemma 7. By Lemma 7, D is a strict fundamental domain. It contains the vertices v_{1}, …, v_{n} in its interior (i.e. outside the boundary ∂D), and thus contains also v_{0} and v_{n+1}. We identify D with the quotient G̃_{v_{1},...,v_{n}}\X. We then look at the projection of δ ∪ δ′ to D. Since the distance in X̃ between v_{0} and v_{n+1} is n + 1, their distance in D is also n + 1 (the distance in the quotient cannot increase, while that in the subcomplex cannot increase). It follows that the length of the projection of δ′ to D is at least n + 1, and so the length of γ is at least 2(n + 1). □

3.6. The fixed point propetry. We now pass to proving Theorem 3.2. The proof will use the following lemma. The proof of a related result, for contractible CW-complexes, can be found in [12]. We remind the reader that “mod-p acyclic” means “having the same mod-p Čech cohomology as a point”.

Lemma 3.34. Fix an integer n > 0, let Y_{1}, …, Y_{n} be closed subspaces of a space X, let Y = ⋃_{i=1}^{n} Y_{i} and let A = ⋂_{i=1}^{n} Y_{i}, the union and intersection of the Y_{i} respectively. Suppose that for all subsets I ⊂ {1, …, n} with 1 ≤ |I| < n, the intersection ⋂_{i∈I} Y_{i} is mod-p acyclic. Then the reduced mod-p Čech cohomologies of Y and A are isomorphic, with a shift in degree of n − 1. More precisely, for each m there is an isomorphism H̃^{m}(Y) ≅ H̃^{m−n+1}(A).

Proof. In the case when n = 1, we have that Y = A and the assertion is trivially true. Now suppose that n ≥ 2. For 1 ≤ i ≤ n − 1, let Z_{i} = Y_{i} ∩ Y_{n}, let Z = ⋃_{i} Z_{i}, and let Y′ = ⋃_{i=1}^{n−1} Y_{i}. By definition, Y − Y′ = Y_{n} − Z, and so by the strong form of excision that holds for Čech cohomology (see the end of section 3.3 of [7]), it follows that H^{∗}(Y, Y′) ≅ H^{∗}(Y_{n}, Z).

By induction on n, we see that for each m, H̃^{m}(Z) ≅ H̃^{m−n+2}(A). Also by induction, we see that Y′ is mod-p acyclic, since ⋂_{i=1}^{n−1} Y_{i} is mod-p acyclic by hypothesis. Since n ≥ 2, the hypotheses also imply that Y_{n} is mod-p acyclic. Hence the long exact sequence in reduced cohomology for the pair (Y_{n}, Z) collapses to isomorphisms, for all i, H̃^{i−1}(Z) ≅ H^{i}(Y_{n}, Z). Similarly, the long exact sequence in reduced cohomology for the pair (Y, Y′) collapses to isomorphisms, for all i, H^{i}(Y, Y′) ≅ H̃^{i}(Y). Putting these isomorphisms together gives an isomorphism, for all i, of reduced cohomology groups H̃^{i−1}(Z) ≅ H̃^{i}(Y). The claimed result follows. □

Proof of Theorem 3.2. Suppose that X is a p-acyclic G-space of finite covering dimension. Let g_{1}, …, g_{n+2} be the elements of S, and let Y_{i} be the points of X that are fixed by g_{i}. By Smith theory, the fixed point set of the action of any finite p-group on X is mod-p acyclic. For an explicit reference, see Theorem III.7.11 of Bredon’s book [3], noting that “finite covering dimension” implies Bredon’s hypothesis “finitistic” (see p. 133 of [3]), and that “X is mod-p acyclic” is equivalent to Bredon’s hypothesis “the pair (X, ∅) is a mod-p Čech cohomology 0-disk”. Hence the subspaces Y_{i} satisfy the hypotheses of Lemma 3.34. The global fixed point set A for the action of G on X is equal to the intersection A = ⋂_{i=1}^{n+2} Y_{i}. If A is not mod-p acyclic, then for some m ≥ −1, the reduced cohomology group H̃^{m}(A) is non-zero. By Lemma 3.34, it follows that the union Y = ⋃_{i=1}^{n+2} Y_{i} has a non-vanishing reduced cohomology group H̃^{j}(Y) for some j ≥ n. Hence the relative cohomology group H^{j+1}(X, Y) is non-zero for some j ≥ n, and so X must have covering dimension at least n + 1. □

4. Finitely presented groups that fail to act

In this section we establish the validity of Template NA_{fp}. We also show the triviality of actions by diffeomorphisms of certain groups, see Lemma 4.5. This immediately implies Proposition 1.3 and provides the input to Template NA_{fp} that is needed to prove Theorem 1.4. Our proof of Lemma 4.5 was inspired by recent work of Bridson and Vogtmann [5].

Definition 4.1. A sequence of groups and monomorphisms, (G_{n}; ξ_{n,j}) (n ∈ ℕ, j = 1, …, J), is called a recursive system if

(i) each G_{n} has a presentation ⟨A_{n} | R_{n}⟩ with A_{n} finite and ⋃_{n} R_{n} ⊂ A^{∗} recursively enumerable, where A = ⊔_{n}A_{n}, and

(ii) each monomorphism ξ_{n,j} : G_{n} → G_{n+1} is defined by a set of words S_{n,j} = {w_{n,j,a} ∈ A_{n+1}^{∗} | a ∈ A_{n}} such that w_{n,j,a} = ξ_{n,j}(a) in G_{n+1}, with ⋃_{n,j} S_{n,j} ⊂ A^{∗} recursively enumerable.

We shall be interested only in sequences where, for each sufficiently large integer n, G_{n+1} is generated by the union of the images of the ξ_{n,j}. And in our applications we shall need only the case J = 2.

Examples 4.2. The following are recursive systems.

(1) Define G_{n} = SL(n, ℤ), set J = 2, and define ξ_{n,1} and ξ_{n,2} to be the embeddings SL(n, ℤ) → SL(n + 1, ℤ) defined by

ξ_{n,1}(M) = ( \begin{matrix}M & 0 \\ 0 & 1 \end{matrix} ) , and ξ_{n,2}(M) = ( \begin{matrix}1 & 0 \\ 0 & M \end{matrix} ) for all M ∈ SL(n, ℤ).

(2) Writing Alt(n) to denote the alternating group consisting of even permutations of n = {1, …, n} and taking J = 2, define G_{n} = Alt(n) and define ξ_{n,1}, ξ_{n,2} : Alt(n) → Alt(n + 1) to be the embeddings induced by the maps I_{n,1}, I_{n,2} : n → n+1 defined by I_{n,1} : k → k and I_{n,2} : k → k + 1.

Lemma 4.3. Let (G_{n}; ξ_{n,j}) (n ∈ ℕ, j = 1, …, J) be a recursive system of non-trivial groups and monomorphisms ξ_{n,j} : G_{n} → G_{n+1}, and suppose that there exists n_{0} so that for each n ≥ n_{0}, G_{n+1} is generated by ⋃_{j} ξ_{n,j}(G_{n}). Then there exists a finitely presented group G_{ω} which for each n ≥ n_{0}, contains two isomorphic copies of G_{n} so that G_{ω} is the normal closure of the union of these two subgroups.

Proof. By renumbering, we may assume that n_{0} = 1. Consider the free product

(2)  G = ∗_{n=1}^{∞} G_{n},

and for each j ∈ J let θ_{j} : G → G be the injective endomorphism of G whose restriction to G_{n} is ξ_{n,j}.

Now, let H be the multiple HNN-extension of G corresponding to these endomorphisms:

H = ⟨G, t_{1}, …, t_{j} ∥ t_{j}gt_{j}^{−1} = θ_{j}(g), g ∈ G, j ∈ J⟩ .

In the notation of Definition 4.1, this has presentation

⟨A, t_{1}, …, t_{j}∥R_{n} (n ∈ ℕ), t_{j}at_{j}^{−1}w_{n,j,a}^{−1} (n ∈ ℕ, j ∈ J, a ∈ A_{n})⟩.

By hypothesis, this is a recursive presentation. Moreover, since the images of the ξ_{n,j} generate G_{n+1}, the group H is generated by the finite set A_{1} ∪ {t_{1}, …, t_{J}}.

Now, by the Higman embedding theorem (see [13, IV.7]), H can be isomorphically embedded into a finitely presented group B. Suppose that B is generated by elements x_{1}, …, x_{l}. Without loss of generality we can and do assume that each of x_{1}, …, x_{l} has infinite order. Indeed, to ensure this one can if necessary replace B by the free product B ∗ ℤ of B with the infinite cyclic group generated by z, which is generated by the elements z, zx_{1}, …, zx_{l} of infinite order. Choose an element of infinite order y ∈ G ≤ B and a subgroup F ≤ G such that F is free of rank l and

(3)  F ∩ ⟨y⟩ = {1}, F ∩ ⟨x_{i}⟩ = {1} for i = 1, …, l.

Such a choice is possible because G is a free product of infinitely many non-trivial groups and a cyclic subgroup of B can intersect at most one free factor non-trivially.

Consider, now, the iterated HNN-extension of B:

L = ⟨B, s_{1}, …, s_{l} ∥ s_{i}x_{i}s_{i}^{−1} = y, i = 1, …, l⟩.

Let {f_{1}, …, f_{l}} be free basis of F. By (3) and Britton’s lemma ([13, IV.2]), the subgroup of L generated by s_{1}, …, s_{l} and F is freely generated by the elements s_{1}, …, s_{l}, f_{1}, …, f_{l}. Let L′ be a copy of L and let s′_{1}, …, s′_{l}, f′_{1}, …, f′_{l} denote the copies of the corresponding elements. Finally we obtain the group that we seek by defining

G_{ω} = ⟨L, L′ ∥ s_{i} = f′_{i}, f_{i} = s′_{i}, i = 1, …, l⟩.

The group G_{ω} is infinite and finitely presented by construction.

A key feature in our construction is that for all k ≤ n the free factor G_{k} of G is conjugate in H (and hence in G_{ω}) to a subgroup of G_{n}, and for k > n the conjugates of G_{n} by positive words of length k − n in the letters t_{j} generate G_{k}. Thus G is the normal closure of each G_{n}. Likewise G′ is the normal closure of G′_{n}. All that remains is to observe that G_{ω} is generated by the set {x_{1}, …, x_{l}, s_{1}, …, s_{l}, x′_{1}, …, x′_{l}, s′_{1}, …, s′_{l}}, each of whose elements is conjugate to an element of G or G′. Thus G_{ω} is the normal closure of G_{n} ∪ G′_{m} for every n, m ≥ 1. □

The following theorem establishes the validity of Template NA_{fp}. Theorem 4.4. If the groups G_{n} satisfy the conditions of Template NA_{fp}, then the finitely presented group G_{ω} constructed in Lemma 4.3 cannot act non-trivially on any X ∈ X. Proof. Suppose that G_{ω} acts on a space X ∈ X. Then X ∈ X_{m} for some m ∈ ℕ and we have a homomorphism α : G_{ω} → Homeo(X) that we want to prove is trivial. By hypothesis, there is some G_{n} that cannot act non-trivially on X. Hence, in the notation of the preceding lemma, α(G_{n}) = α(G′_{n}) = {1}. Therefore the kernel of α is the whole of G_{ω}. □ Lemma 4.5. If G is a simple group that contains a copy of ℤ_{p}^{n+1}, then G cannot act effectively by diffeomorphisms on any smooth p-acyclic manifold X of dimension at most n. Proof. Let E be the subgroup of G isomorphic to (ℤ_{p})^{n+1}. Fix a Riemannian metric on X that is compatible with the smooth structure and, given an action of G, average the metric over the action of E to ensure that the action of E is by isometries.

Smith theory tells us that the action of E on X has mod-p acyclic fixed point set (Theorem III.7.11 of [3]), and so there is a point x fixed by E. Taking derivatives at x, we obtain an action of E on the tangent space T_{x}(X). But E has no faithful linear representations of dimension less than or equal to n, and so some non-identity element h ∈ E acts trivially on T_{x}(X). Since the action of E is by isometries, the exponential map is E-equivariant and so h acts trivially on an open ball about x. But the fixed point set for any isometry of X is a closed submanifold, and so it follows that h fixes the whole of X.

Thus h lies in the kernel of the map G → Diffeo(X). Since G is simple, it follows that G acts trivially. □

Proof of Theorem 1.4 The preceding lemma tells us that, given any prime p and positive integer n, any alternating group Alt(m) with m sufficiently large cannot act non-trivially by diffeomorphisms on a smooth p-acyclic manifold of dimension less than n. It follows that the group G_{ω} obtained by applying Theorem 4.4 to the recursive system in Example 4.2(2) cannot act non-trivially by diffeomorphisms on a p-acyclic manifold of any dimension, for all primes p.

Remark 4.6. (1) For each fixed prime p, C. Röver constructed a finitely presented simple group containing, for each n, a copy of (ℤ_{p})^{n} [20]. By Lemma 4.5, such a group cannot act non-trivially by diffeomorphisms on any mod-p acyclic manifold.

(2) It should be clear from the architecture of our proof how suitable variations on Lemma 4.5 will give rise to analogues of Proposition 1.3 and Theorem 1.4. The work of Bridson and Vogtmann [5] provides several such variations.

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(G. Arzhantseva) Université de Genève, Section de Mathématiques, 2-4 rue du Lièvre, Case postale 64, 1211 Genève 4, Switzerland

E-mail address: Goulnara.Arjantseva@math.unige.ch

(M.R. Bridson) Mathematical Institute, 24-29 St Giles’, Oxford, UK

E-mail address: bridson@maths.ox.ac.uk

(T. Januszkiewicz) Department of Mathematics, The Ohio State University, 231 W. 18th Ave., Columbus, OH 43210, USA and The Mathematical Instutute of Polish Academy of Sciences. On leave from Instytut Matematyczny, Uniwersytet Wrocławski

E-mail address: tjan@math.ohio-state.edu

(I. J. Leary) Department of Mathematics, The Ohio State University, 231 W. 18th Ave., Columbus, OH 43210, USA

E-mail address: leary@math.ohio-state.edu

(A. Minasyan) Université de Genève, Section de Mathématiques, 2-4 rue du Lièvre, Case postale 64, 1211 Genève 4, Switzerland. On leave from School of Mathematics, University of Southampton, Highfield, Southampton, SO17 1BJ, United Kingdom.

E-mail address: aminasyan@gmail.com

(J. Świa̧tkowski) Instytut Matematyczny, Uniwersytet Wroclawski, pl. Grunwaldzki 2/4, 50-384 Wroclaw, Poland

E-mail address: swiatkow@math.uni.wroc.pl