Some groups of type VF

Ian J. Leary and Brita E. A. Nucinkis

Abstract A group is of type V F if it has a finite-index subgroup which has a finite classifying space. We construct groups of type V F in which the centralizers of some elements of finite order are not of type V F and groups of type V F containing infinitely many conjugacy classes of finite subgroups. It follows that a group G of type V F need not admit a finite-type universal proper G-space. We construct groups G for which the minimal dimension of a universal proper G-space is strictly greater than the virtual cohomological dimension of G. Each of our groups embeds in GL_{m}(ℤ) for sufficiently large m. Some applications to K-theory are also considered.

1. INTRODUCTION

In [6], M. Bestvina and N. Brady gave a method for constructing torsion-free groups with given homological finiteness properties. Examples that they constructed include: for each n ≥ 0, groups of type F_{n} that are not of type F_{n+1}, groups that are of type FP over certain rings but not FP over others, and groups that are FP over the integers but that are not finitely presentable. (These finiteness conditions will be defined below.) We show how to construct virtually torsion-free groups with surprising homological properties as finite extensions of the groups considered by M. Bestvina and N. Brady. (By a finite extension of a group H we mean a group G containing H as a finite-index normal subgroup.)

The strongest of the homological finiteness conditions is F, where a group H is of type F if there is a finite Eilenberg-Mac Lane space K(H, 1). Groups of type F are necessarily torsion-free. We exhibit groups that are virtually of type F or ‘of type VF’ that contain infinitely many conjugacy classes of finite subgroups, and groups of type VF in which the centralizers of some elements of finite order are not of type VF. The question of whether such examples can exist is implicit in [13,14,18,1], and was posed explicitly in the problem list from the 1977 Durham Symposium edited by C. T. C. Wall [58], and by W. Lück in [38]. One reason why these examples are of interest is that they show that the property of having a finite (resp. finitely dominated, finite type) classifying space for proper actions is not preserved under finite extensions. (The classifying space for proper actions of G is also known as the universal proper G-space, and usually denoted by \underline{E}G.) A result due to K. S. Brown ([15] IX.13.2) implies that any group of type VF contains only finitely many conjugacy classes of subgroups of prime power order. Our examples include groups of type VF containing infinitely many conjugacy classes of finite abelian subgroups.

For each of the groups that we construct there is a ‘nice’ finite-dimensional (but not finite) model for the classifying space for proper actions. In certain cases a study of this model gives rise to a lower bound for the dimension of any model which is strictly greater than the virtual cohomological dimension. In particular, for any n ≥ 1 we exhibit a group G of type VF with virtual cohomological dimension 3n and such that any model for \underline{E}G has dimension at least 4n. These groups answer a question posed by K. S. Brown [14,38,43].

Using the results of [21], presentations may be given for the groups that we construct, and we give some examples including an explicit presentation for a group of type VF that contains infinitely many conjugacy classes of subgroup of order 120. Moreover, this group has virtual cohomological dimension three and any model for its classifying space for proper actions has dimension at least four. Each of the groups can be embedded in the integral general linear group GL_{n}(ℤ) for sufficiently large n. For example, the group mentioned above embeds in GL_{46}(ℤ). This group answers a question of M. Bridson [10].

Our method can also be used to construct groups satisfying weaker finiteness conditions than VF. For example, we exhibit a finitely presented subgroup of GL_{6}(ℤ) containing infinitely many conjugacy classes of elements of order two, and for each n ≥ 4 a finitely presented subgroup of GL_{4n}(ℤ) containing infinitely many conjugacy classes of elements of order m, for each m > 1 dividing n. These groups answer a question posed by F. Grunewald and V. Platonov [25,48]. (M. Bridson has also answered this question, in the same article in which he posed the variant mentioned above [10].)

Modulo torsion, the Farrell-Jones conjectures and the Baum-Connes conjecture relate the homology of centralizers of elements of finite order in the group G to the K-theory of various algebras associated to G [5,20,37]. The Baum-Connes conjecture may be shown to hold for the groups that we construct. Using this fact we exhibit groups of type VF for which the higher algebraic K-groups of certain group algebras and the topological K-groups of the reduced C^{∗}-algebra have infinite rank.

The input to the Bestvina-Brady construction is a finite flag complex L, and the output is a torsion-free group H_{L} whose homological finiteness properties are controlled by the homotopy type of L. For example, H_{L} is FP over the integers but not finitely presentable if and only if L is acyclic but not contractible. The input for our generalization consists of a finite flag complex L, together with a group Q of simplicial automorphisms of L, and the output is a group H̃ = H_{L} ⋊ Q whose homological properties are controlled by the homotopy type of L and of L^{P}, the P -fixed points in L, for each P ≤ Q. For example, H̃ is of type VF if and only if L is contractible, and H̃ contains infinitely many conjugacy classes of finite subgroups if and only if Q acts on L without a global fixed point.

To construct examples, we rely on various results concerning actions of finite groups on simplicial complexes, including work of R. Oliver and earlier work that is surveyed by him in [46]. The Bestvina-Brady groups H_{L} are subgroups of right-angled Artin groups (or graph groups), and we also use a result of J. S. Crisp describing the fixed subgroup for a group of symmetries of an Artin group [19]. In fact, we only need a simple special case of Crisp’s theorem, for which we provide a proof. Our proof that the groups H̃ are linear over ℤ uses a result due to T. Hsu and D. Wise [29]. An argument due to G. A. Niblo and L. Reeves shows that the groups H̃ lie in a class of groups for which N. Higson and G. Kasparov have shown that the Baum-Connes conjecture holds [44,26,31]. Work of M. Matthey and W. Lück can then be used to relate the rational homology of centralizers of elements of finite order in H̃ to various K-groups [40,41,37].

The remainder of the paper is organized as follows. Section 2 recalls the definitions and properties of the finiteness conditions that will be used. Section 3 contains a statement of the Bestvina-Brady theorem, together with some information concerning its proof which is required in Sections 4–6. Section 4 describes our construction, and contains our main theorem, Theorem 3. Section 5 defines classifying spaces for proper actions, and explains their connection to our work. Section 6 compares various algebraic dimensions for a group with the dimension of its classifying space for proper actions. In Section 8 presentations and faithful integral linear representations for our groups are given. Sections 7 and 9 describe our examples and some of the questions that they resolve. The examples in Section 9 are rather more explicit than those in Section 7. Section 7 also discusses some related questions which our method fails to resolve, notably a conjecture of H. Bass concerning groups of type FP over the rationals [4]. Finally, in Section 10 partial information concerning the rational homology of Bestvina-Brady groups is obtained and used to construct group algebras whose higher algebraic K-groups have infinite rank.

While working on this project the authors met both in Southampton and in Zürich, and the first named author visited Münster and Barcelona. The authors gratefully acknowledge the support and hospitality of each other’s institutions, and of the CRM in Barcelona and the SFB Mathematik (478) in Münster. The authors also thank Noel Brady, John Crisp, Warren Dicks and Michel Matthey for helpful comments. Finally, they thank Wolfgang Lück for helpful conversations and for encouraging them to study the dimension of the classifying space for proper actions.

2. FINITENESS CONDITIONS

Let G be a discrete group, let R be a commutative ring, and let RG denote the group algebra of G. The group G is of type F (resp. type F_{n}) if there is an Eilenberg-Mac Lane space K(G, 1) which is finite (resp. has finite n-skeleton). It is easily seen that every group G is F_{0}, that G is F_{1} if and only if G can be finitely generated, and that G is F_{2} if and only if G can be finitely presented. The ring R may be made into a module for RG by letting every element of G act as the identity. This module is known as the trivial module for RG. The group G is FP over R if the trivial module admits a finite resolution by finitely generated projective RG-modules, i.e., for some n and finitely generated projective modules P_{0}, …, P_{n}, there is an exact sequence

0 → P_{n} → P_{n−1} → ⋯ → P_{1} → P_{0} → R → 0.

If the P_{i} may be chosen to be finitely generated free RG-modules then G is FL over R.

If the trivial module admits a projective resolution (of any length) in which the first n terms are finitely generated, then G is of type FP_{n} over R. Equivalently, there are finitely generated projective modules P_{0}, …, P_{n} and an exact sequence

P_{n} → P_{n−1} → ⋯ → P_{1} → P_{0} → R → 0.

G is of type F_{∞} (resp. FP_{∞} over R) if, for all n ≥ 0, G is of type F_{n} (resp. FP_{n} over R).

If X is any of the above finiteness conditions, a group G is said to be of type VX or ‘virtually of type X’ if G has a finite-index subgroup H of type X. Many of the virtual notions are redundant, since for each n and R, VF_{n} is equivalent to F_{n}, and VFP_{n} over R is equivalent to FP_{n} over R. A theorem due to Serre shows that if G is VFP over R and contains no R-torsion, then G is FP over R [7,15,51]. In general, if G is VF (resp. VFP over R) then G is F_{∞} (resp. FP_{∞} over R). Note that the R. Thompson group is of type F_{∞}, but is not VF or VFP over any non-trivial R since it contains a free abelian group of infinite rank [16].

3. THE BESTVINA-BRADY CONSTRUCTION

Let L be a simplicial complex. L is a flag complex (or full simplicial complex) if there is no larger simplicial complex with the same 1-skeleton. Given a simplicial graph L^{1}, one can recover a flag complex L as the clique complex of L^{1}, i.e., the complex whose n-simplices are the complete subgraphs of L^{1} on (n+1) vertices. The barycentric subdivision of any simplicial complex is a flag complex.

Given a finite flag complex L, the right-angled Artin group (or graph group) G_{L} is a group whose generators are in 1-1 correspondence with the vertex set of L, subject only to the relations that the ends of every edge of L commute with each other. For example, in the case when L is the n-simplex, G_{L} is free abelian of rank n + 1, and in the case when L consists of n vertices and no edges, G_{L} is free of rank n. Provided that L is not empty, there is a surjective homomorphism ϕ_{L} : G → ℤ that sends each of the given generators to 1. The group H_{L} is defined to be the kernel of ϕ_{L}. (For example, in the case when L is the n-simplex, H_{L} is free abelian of rank n, and in the case when L consists of n > 1 points, H_{L} is free of infinite rank.)

Before stating the Bestvina-Brady theorem, we recall some topological definitions. A topological space X is 0-connected if and only if X is non-empty and connected. For n ≥ 1, X is n-connected if and only if X is 0-connected and for each i with 1 ≤ i ≤ n, the homotopy group π_{i}(X) is trivial. X is R-homologically n-connected if H_{i}(X; R) is isomorphic to the ith homology group of a point for all i with 0 ≤ i ≤ n, and X is R-acyclic if X is R-homologically n-connected for all n.

Theorem 1 [6]. Let L be a non-empty finite flag complex, let n ≥ 0, and define H = H_{L} as above. H is of type F if and only if L is contractible; H is of type F_{n+1} if and only if L is n-connected; H is of type FP over R if and only if L is R-acyclic; H is of type FP_{n+1} over R if and only if L is R-homologically n-connected.

The proof of this theorem uses nice models for an Eilenberg-Mac Lane space K(G, 1) for G = G_{L}, and for its universal cover, EG. The model for K(G, 1) has one 0-cell, and for n ≥ 0 its n-cells are n-dimensional cubes in bijective correspondence with the (n − 1)-simplices of L. The corresponding model for EG will be denoted by X = X_{L}. Thus X is a CW-complex whose n-cells are cubes indexed by pairs (gx_{0}, {v_{1}, …, v_{n}}), where x_{0} is some fixed vertex of X, g ∈ G and v_{1}, …, v_{n} are 0-simplices of L that span an (n − 1)-simplex (or equivalently, distinct elements of the given generating set for G that all commute). For each i, the (n − 1)-cubes (gx_{0}, {v_{1}, …, v_{i−1}, v_{i+1}, … v_{n}}) and (gv_{i}x_{0}, {v_{1}, …, v_{i−1}, v_{i+1} … v_{n}}) are a pair of opposite faces of the n-cube given above. The action of G on X is given by h.(gx_{0}, σ) = (hgx_{0}, σ).

There is a CAT(0)-metric on X in which each of the cubes is isometric to a standard Euclidean cube of side length 1. This implies that X is contractible, and gives one method for proving that X/G is a K(G, 1). Make the real line ℝ into a CW-complex with 0-cells the integers, let ℤ act on ℝ by addition, and use ϕ_{L} : G_{L} → ℤ to define an action of G on ℝ in which H acts trivially. Then there is a G-equivariant map f: X → ℝ which is cellular, affine on each cube, and for each g ∈ G and v ∈ L^{0} sends the edge from gx_{0} to gvx_{0} to an interval [m, m + 1]. It is possible to choose f so that f(x_{0}) = 0, in which case m = ϕ_{L}(g). For t ∈ ℝ, define X_{t} = f^{−1}(t). The cellular structure on X gives rise to a cellular structure on X_{t}, but note that X_{t} intersects the cubes of X ‘diagonally’, so X_{t} is not a cube complex. For each t, H is the setwise stabilizer in G of X_{t}. Since X is a model for EG, it is also an infinite model for EH. On the other hand X_{t} is a finite H-CW-complex, but in general is not contractible. The proof of the theorem involves comparing the homotopy types of X and X_{t} using a ‘cellular Morse-theory’. In this way it is shown that X is homotopy equivalent to a space made by attaching infinitely many cones on L to X_{t}.

4. FINITE EXTENSIONS

The map L ↦ G_{L} is a functor from flag complexes to groups, and in particular any automorphism of L induces an automorphism of G_{L}. Let Q be a group of automorphisms of L, and suppose that Q acts simplicially on L, in the sense that for every simplex σ of L, the setwise and pointwise stabilizers of σ are equal. For an arbitrary action of a finite group Q on a simplicial complex L, the induced action of Q on the barycentric subdivision L′ is a simplicial action on a flag complex, so insisting that the action should be simplicial places no restriction on the equivariant homotopy type of L. The hypothesis that the action of Q on L be simplicial ensures that the fixed points L^{Q} form a subcomplex of L. The following theorem is a special case of a theorem of J. S. Crisp [19].

Theorem 2 [19]. Let L be a finite flag complex, and suppose that Q acts simplicially on L, with fixed points M = L^{Q}. Then the subgroup of G_{L} fixed by Q is isomorphic to G_{M}.

Proof. Given a choice of base-vertex x_{0} for X, the action of Q on L and on G = G_{L} induces an action of Q by isometries on X = X_{L}, fixing x_{0}. The fact that X is CAT(0) implies that the fixed point subset X^{Q} is contractible ([9] or [11]). The fact that Q acts simplicially on L implies that the action of Q on X is cellular, and hence that X^{Q} is a subcomplex of X.

The vertices of X^{Q} are precisely the vertices gx_{0} such that g ∈ G^{Q}. Since X^{Q} is a connected subcomplex of X, any two vertices of X^{Q} must be joined by a path in the 1-skeleton of X^{Q}. But the edges of X^{Q} are the edges (gx_{0}, gvx_{0}) such that v ∈ M^{0} and g ∈ G^{Q}, and so the natural map from G_{M} to (G_{L})^{Q} is surjective. The kernel, K, of this map is trivial, since the map induces a homeomorphism from X_{M}/K to X^{Q}, and both X_{M} and X^{Q} are contractible.

Remark. Crisp’s theorem is far more general. As well as applying in the case when the action of Q on L is not simplicial, it also applies to other Artin groups (not right-angled) for which a suitable CAT(0) space exists [19].

For any finite group Q acting on a non-empty flag complex L, the homomorphism ϕ_{L} : G_{L} → ℤ is Q-equivariant (for the trivial action on ℤ). It follows that Q acts on H_{L}, and we may form a group H̃ = H̃(L, Q) = H_{L} ⋊ Q, the split extension with kernel H_{L} and quotient Q. The homomorphism ϕ_{L} extends uniquely to a homomorphism, which we shall also denote by ϕ_{L}, from G̃ = G_{L} ⋊ Q to ℤ, and H̃ is equal to the kernel of this homomorphism. Since H_{L} is torsion-free and normal in H̃, any finite subgroup of H̃ will map isomorphically to a subgroup of Q.

Theorem 3. Let L be a non-empty finite flag complex with an effective simplicial action of a finite group Q, and define H̃ = H̃(L, Q) as above. Let P be a subgroup of Q, let M = L^{P}, and let P be the set of subgroups of H̃ that map isomorphically to P or one of its conjugates in Q.

1. If M = ∅, then P contains infinitely many conjugacy classes of subgroups of H̃, and for all P′ ∈ P,

C_{H}(P′) ≅ C_{Q}(P).

2. If M ≠ ∅, then all elements of P are conjugate in H̃, and for any P′ ∈ P, C_{H}(P′) ≅ H̃(M, C_{Q}(P)) = H_{M} ⋊ C_{Q}(P).

Proof. A choice of base-vertex x_{0} for X defines an action of Q on X, and the actions of G_{L} and Q on X together give a proper action of the split extension G̃ = G_{L} ⋊ Q. For this action the stabilizer of the vertex gx_{0} is the subgroup gQg^{−1}, where by an abuse of notation we are using Q to denote the ‘standard’ copy of Q inside G̃. The G-equivariant map f : X → ℝ defined during the description of the proof of Theorem 1 is also G̃-equivariant, where G̃ acts on ℝ by the translation action of ℤ = ϕ_{L}(G̃).

Since X is CAT(0) and G̃ is acting by isometries, it follows that, for any finite subgroup P′ of G̃, the fixed points X^{P′} form a contractible subcomplex of X. In particular, X^{P′} contains some vertex gx_{0}, and so P′ ≤ gQg^{−1}. If P′ and P′′ are subgroups of G̃ that both map isomorphically to P ≤ Q, it follows that P′ and P′′ are conjugate in G̃.

In the case when M is empty, X^{P′} cannot contain any 1-cells, so must be a single point gx_{0}. In this case, f(X^{P′}) is the point ϕ_{L}(g) ∈ ℤ ≤ ℝ. If P′′ = hP′h^{−1}, then f(X^{P′′}) is the point ϕ_{L}(hg). But if h ∈ H̃, then ϕ_{L}(hg) = ϕ_{L}(g), and so H̃-conjugate subgroups isomorphic to P fix points of X of the same ‘height’. It follows that in the case when M is empty, the subgroups v^{i}P′v^{−i} for i ∈ ℤ all lie in distinct conjugacy classes in H̃, where v is any of the standard generators for G_{L}.

In the case when M is not empty, let P′ be a subgroup of H̃ that maps isomorphically to P such that P′ fixes x_{0}. If P′′ is another subgroup of H̃ that maps isomorphically to P, then there exists g ∈ G_{L} such that P′′ fixes the point gx_{0}. Since M is non-empty, there exists v, one of the generators for G_{L}, which commutes with P′. Then for some i ∈ ℤ, the element g′ = gv^{i} lies in H̃ and satisfies g′P′g^{′−1} = gv^{i}P′v^{−i}g^{−1} = gP′g^{−1} = P′′.

The description of the centralizers in each case follows from Theorem 2.

Remark. (Non-simplicial actions.) A similar theorem can be proved in the more general case, in which Q is an arbitrary group of automorphisms of the flag complex L. The general version of Crisp’s theorem (not Theorem 2) implies that the subgroup of G_{L} fixed by P ≤ Q is isomorphic to G_{M}, where M is a flag complex whose realization |M| is homeomorphic to |L|^{P}, the P -fixed points in the realization of L. The vertices of M are the barycentres of simplices of L that are minimal amongst simplices that are preserved (setwise) by P, and simplices of M are sets of vertices that are contained within a single simplex of L. The inclusion of G_{M} in G_{L} sends the generator corresponding to the simplex (v_{1}, …, v_{n}) to the product v_{1} ⋯ v_{n}, and so the composite of the inclusion with ϕ_{L} is not equal to ϕ_{M}, but to a homomorphism ϕ′ : G_{M} → ℤ that sends each generator to some strictly positive integer. J. Meier, H. Meinert and L. VanWyk have given a version of the Bestvina-Brady theorem that applies also to homomorphisms such as ϕ′ [42].

In this way, one obtains the same statement as Theorem 3 for the case when Q does not act simplicially (with the new definition of M), except that if P ≤ Q fixes some point of |L| but does not fix a vertex, the set P may contain more than one (but still only finitely many) conjugacy classes of subgroup of H̃. In fact, the number of conjugacy classes in P is equal to the highest common factor of the numbers of vertices of the minimal simplices that are preserved by P, or equivalently the index of ϕ_{L}(G_{M}) in ℤ. This phenomenon can be seen already in the simplest possible case, namely Q = Σ_{n}, the symmetric group of degree n, acting as the full automorphism group of the (n − 1)-simplex L. In this case G̃ is the wreath product ℤ ≀ Σ_{n} and H̃ is an extension with kernel H_{L} = ℤ^{n−1} and quotient Σ_{n}, which contains n conjugacy classes of subgroup isomorphic to Σ_{n}. In this special case, this can also be verified directly by computing H^{1}(Σ_{n}; H_{L}).

5. CLASSIFYING SPACES FOR PROPER ACTIONS

For G a discrete group, a proper G-CW-complex is one in which all cell stabilizers are finite. A classifying space for proper actions of G, denoted by \underline{E}G, is a proper G-CW-complex such that for each finite P ≤ G, the P -fixed point set (\underline{E}G)^{P} is contractible. Equivalently, \underline{E}G is a terminal object in the homotopy category of proper G-CW-complexes. From the second definition, it follows that \underline{E}G is unique up to equivariant homotopy. For more details, including a proof that these definitions are equivalent and a construction of \underline{E}G for general G, see any of [56,57,34,43]. The definitions of \underline{E}G given above may be compared with those for EG, which is defined as either the universal cover of a K(G, 1), or as a contractible free G-CW-complex, or as a terminal object in the homotopy category of free G-CW-complexes. In particular, if G is torsion-free, there is no distinction between free and proper actions, and so \underline{E}G = EG.

Just as the finiteness conditions F, F_{n} and F_{∞} may be viewed as conditions on the number of G-orbits of cells in a model for EG, one may consider finiteness conditions for \underline{E}G. For example, every finite subgroup of G fixes a point of \underline{E}G, and so must be a subgroup of a 0-cell stabilizer. Hence the existence of an \underline{E}G with only finitely many orbits of 0-cells implies that G contains only finitely many conjugacy classes of finite subgroups. If P is a finite subgroup of G with normalizer N = N_{G}(P), then it may be shown that (\underline{E}G)^{P} is a model for \underline{E}N, which is finite (resp. finitely dominated, resp. of finite type) if the original \underline{E}G had this same property. In particular, if there is an \underline{E}G that is finite (resp. finitely dominated, resp. of finite type), then any torsion-free finite-index subgroup of any N_{G}(P) must be of type F (resp. FP, resp. F_{∞}). In fact, it may be shown that conditions on finite subgroups and their normalizers characterize those G for which \underline{E}G is finitely dominated (resp. finite type). The following theorem is a combination of theorem 1 of F. Connolly and T. Koźniewski’s [18] and theorem 4.2 of W. Lück’s [38]. (Note however that we have modified Lück’s statement, which uses N_{G}(P)/P instead of N_{G}(P).)

Theorem 4. For any discrete group G, the following are equivalent:

1. There is an \underline{E}G of finite type;

2. G contains finitely many conjugacy classes of finite subgroups, and for each such P, N_{G}(P) is of type F_{∞}.

If G is of finite virtual cohomological dimension and satisfies condition 1 or 2, then \underline{E}G is finitely dominated.

This result indicates that Theorem 3 may be used to construct groups of type VF that do not have finite type models for \underline{E}G. However, more can be said. Any proper G-CW-complex that is equipped with a G-equivariant CAT(0)-metric is a model for \underline{E}G (see [11,9], but note that no explicit mention of \underline{E}G is made in [11]). In particular, for any finite flag complex L and group Q of simplicial automorphisms of L, the space X_{L} is a model for \underline{E}G̃ = \underline{E}(G_{L} ⋊ Q). The extension of the Bestvina-Brady argument to this context gives:

Theorem 5. With notation as in Theorem 3, the following are equivalent:

1. For each P ≤ Q, L^{P} is contractible;

2. For any t ∈ ℝ, X_{t} is a finite model for \underline{E}H̃;

3. There is a finite model for \underline{E}H̃;

4. \underline{E}H̃ is finitely dominated;

5. There is a finite type model for \underline{E}H̃.

Proof. 2 ⇒ 3, 3 ⇒ 4 and 3 ⇒ 5 are clear. Bestvina and Brady prove 1 ⇒ 2 in the case when Q is trivial using the following lemma. For any L, the space X_{L} (which is an infinite model for EH_{L}) is homotopy equivalent to a CW-complex made by attaching infinitely many copies of the cone on L to X_{t}. The same argument shows that for any finite P′ ≤ G̃, either (X_{t})^{P′} = ∅, or X^{P′} is homotopy equivalent to (X_{t})^{P′} with infinitely many copies of the cone on M = L^{P} attached. The case when (X_{t})^{P′} = ∅ can only occur if M = ∅. In particular, if each such M is contractible, then each (X_{t})^{P′} is homotopy equivalent to X^{P′}, which is contractible, and so X_{t} is a finite model for \underline{E}H̃.

If 1 does not hold, then either (a) there exists P ≤ Q for which L^{P} = ∅, or (b) there exists P ≤ Q and n ≥ 0 for which M = L^{P} is (n − 1)-connected but not n-connected. (Here we adopt the convention that ‘−1-connected’ means ‘non-empty’.) By Theorem 3, in case (a) H̃ contains infinitely many conjugacy classes of finite subgroups and so any model for \underline{E}H̃ contains infinitely many orbits of 0-cells. In case (b), let P′ ≤ H̃ be a subgroup that maps isomorphically to P ≤ Q, and let M = L^{P}. By Theorem 3, the normalizer in H̃ of P′ contains the torsion-free group H_{M} as a finite-index subgroup. By Theorem 1, H_{M} is type F_{n} but not type F_{n+1}, and so any model for \underline{E}H̃ must contain infinitely many orbits of (n + 1)-cells (whose stabilizers contain a conjugate of P′). In either case, if 1 does not hold, then neither can 4 or 5.

Note that the use of an explicit model for \underline{E}H̃ allows us to avoid using the ‘difficult’ implication in Theorem 4 while proving Theorem 5. Note also that condition 1 in Theorem 5 is equivalent to the statement that L is a model for \underline{E}Q.

6. NOTIONS OF DIMENSION

The minimal dimension of a model for \underline{E}G, which we shall denote by \underline{gd}G and refer to as the ‘proper geometric dimension of G’, is a geometrically-defined invariant of G. The relationship between \underline{gd}G and various algebraically-defined quantities, including the virtual cohomological dimension of G or vcdG, has attracted some interest. In this section, we shall give some information concerning \underline{gd}G and these algebraic quantities in the case when G = H_{L} ⋊ Q. This will enable us to construct examples of groups (of type VF) for which \underline{gd}G is strictly greater than vcdG. First we recall the definitions of various algebraic quantities. For more details concerning these definitions we refer the reader to [9].

Recall that the cohomological dimension of a group H or cdH, is defined to be the projective dimension of ℤ as a ℤH-module. (Note that cdH, and all the other notions of dimension that we shall discuss may be either a positive integer or infinity.) Recall also that the virtual cohomological dimension of a group G is by definition vcdG = cdH if G contains a finite-index torsion-free subgroup H, and vcdG = ∞ if G contains no such subgroup. (A lemma due to Serre shows that this is independent of choice of H [15,51].) If H is a torsion-free subgroup of G, then any model for \underline{E}G will serve as a model for EH, and so whenever G is virtually torsion-free, vcdG ≤ \underline{gd}G. Note that there are groups G which are not virtually torsion-free but admit finite-dimensional models for \underline{E}G [9,50]. The question of whether vcdG = \underline{gd}G for all virtually torsion-free groups was raised by K. S. Brown in [14], p32, but see also [38] problem 7.10 and [43].

Another algebraic dimension for a group G is F-cdG, which is defined in terms of relative homological algebra, relative to the class F of all finite subgroups of G. More precisely, F-cdG is defined to be the minimal length n of an exact sequence of G-modules

0 → C_{n} → C_{n−1} → ⋯ → C_{0} → ℤ → 0,

with the properties that

(a) each C_{i} is a direct summand of a sum of modules induced up from finite subgroups of G;

(b) for each finite subgroup H ≤ G, there is an H-equivariant splitting of the exact sequence. A theorem due independently to S. Bouc and to P. H. Kropholler and C. T. C. Wall [8,33] implies that

the augmented cellular chain complex for any acyclic proper G-CW-complex is such an exact sequence. In particular, for any G one has that F-cdG ≤ \underline{gd}G.

The closest algebraic approximation to \underline{gd}G however is another quantity, \underline{cd}G, defined in terms of the category of ‘Bredon coefficient systems’ or ‘modules over the orbit category O(F, G)’. Work of M. Dunwoody (for the 1-dimensional case) and W. Lück (for the high-dimensional case) shows that \underline{cd}G = \underline{gd}G, except that possibly there may exist a group for which \underline{cd}G = 2 and \underline{gd}G = 3 [23,35]. In [9] Noel Brady and the current authors exhibit groups for which strict inequality holds.

Briefly, the orbit category O(F, G) has as objects the G-sets G/P for P a finite subgroup of G, and morphisms the G-maps. An O(F, G)-module or Bredon coefficient system for proper G-CW-complexes is a contravariant functor from O(F, G) to abelian groups. Examples of O(F, G)-modules include a constant functor Z taking each object to ℤ and each morphism to the identity, and for each finite Q ≤ G a functor P_{G/Q} such that P_{G/Q}(G/P) is the free abelian group with basis the morphisms in O(F, G) from G/P to G/Q. Each P_{G/Q} is projective, and any O(F, G)-module is a homomorphic image of a direct sum of such modules. It follows that there are enough projectives in the category of O(F, G)-modules. Thus \underline{cd}G may be defined to be the projective dimension in this category of the object Z. For any proper G-CW-complex X, the map G/P ↦ Map_{G}(G/P, X) describes a functor from O(F, G) to CW-complexes, and the composite of this with the cellular chain complex functor gives rise to a chain complex of projective O(F, G)-modules. In particular, any model for \underline{E}G of dimension n gives rise to an O(F, G)-projective resolution of Z of length n, and so \underline{cd}G ≤ \underline{gd}G.

For more details concerning any of these definitions see [9], where the following inequalities are also proved: for any group G, one has that

F-cdG ≤ \underline{cd}G ≤ \underline{gd}G ≤ max\{\underline{cd}G, 3\},

and that vcdG ≤ F-cdG provided that G is virtually torsion-free. To establish the existence of virtually torsion-free groups G for which vcdG = F-cdG < \underline{cd}G = \underline{gd}G, we shall use the cohomology of \underline{E}G relative to its singular set to provide a lower bound for \underline{cd}G. If X is a proper G-CW-complex, denote by X^{sing} the subcomplex consisting of all cells whose stabilizer is non-trivial. (In [38], W. Lück uses the notation σ_{G}(X) for X^{sing} to emphasize that X^{sing} depends on G as well as on X.) If Y is another proper G-CW-complex, then any equivariant map from X to Y sends X^{sing} to Y^{sing}, and any equivariant homotopy from X × I to Y restricts to an equivariant homotopy from X^{sing} × I to Y^{sing}. In particular, the pair (\underline{E}G, \underline{E}G^{sing}) is welldefined up to equivariant homotopy, and the equivariant cohomology groups H_{G}^{∗}(\underline{E}G, \underline{E}G^{sing}; ℤG) depend only on G. These groups provide a lower bound on the dimension of \underline{E}G. Moreover, it may be checked that

H_{G}^{∗}(\underline{E}G, \underline{E}G^{sing}; ℤG) ≅ Ext_{O(F,G)}^{∗}(Z, P_{G/1}),

and so these groups also provide a lower bound for \underline{cd}G.

Now suppose that L is an n-dimensional flag complex with a simplicial action of the finite group Q. In this case G = H_{L} ⋊ Q admits an n + 1-dimensional model for \underline{E}G, namely the complex X_{L}, and H_{L} contains free abelian subgroups of rank n (since G_{L} contains free abelian subgroups of rank n + 1). It follows that each of the notions of dimension defined in this section is equal to either n or n + 1 for any such G. Ideally, one would like a result determining these quantities for all pairs (L, Q). However, the following theorem will suffice for constructing examples.

Theorem 6. Let L be an n-dimensional acyclic flag complex with a simplicial action of a finite group Q. Suppose that there exists an n-simplex σ of L whose stabilizer in Q is trivial, but such that the stabilizer of each simplex in the boundary of σ is non-trivial. For any m ≥ 1, let G = G(m) be the direct product of m copies of the group H_{L} ⋊ Q. Then G is type VFP over ℤ, and

vcdG = F-cdG = m.n, \underline{cd}G = \underline{gd}G = m.(n + 1).

If L is contractible, then G is of type VF.

Proof. The facts that G is VFP over ℤ and is VF when L is contractible follow from Theorem 1. Since H_{L} contains a free abelian subgroup of rank n, it follows that G contains a free abelian subgroup of rank m.n, which gives the required lower bound for vcdG. For any t, the level set X_{t} is an acyclic n-dimensional space admitting a proper cellular action of H_{L} ⋊ Q. Hence the direct product (X_{t})^{m} is an m.n dimensional acyclic G-CW-complex, and so by a theorem of Bouc and Kropholler-Wall [8,33], F-cdG ≤ m.n.

The direct product X^{m} = (X_{L})^{m} is an m.(n + 1)-dimensional model for \underline{E}G, and so \underline{gd}G ≤ m.(n + 1). Recall the explicit description of the cellular structure of X = X_{L} given after the statement of Theorem 1, and note that since each cell of X is a cube, the product of cells gives rise to a cubical cell structure on X^{m}. For the action of H_{L} ⋊ Q on X, the stabilizer of the k-dimensional cube (gx_{0}, {v_{1}, …, v_{k}}) is a subgroup isomorphic to the stabilizer in Q of the (k −1)-simplex (v_{1}, …, v_{k}). Under the hypotheses on L and Q, there is an (n + 1)-cube in X lying in a free H_{L} ⋊ Q-orbit, such that each of its proper faces lies in a non-free H_{L} ⋊ Q-orbit. Similarly, the direct product X^{m} contains an m.(n + 1)-cube in a free G-orbit, each of whose proper faces lies in a non-free G-orbit. It follows that the relative cellular chain complex C_{∗}(X^{m}, (X^{m})^{sing}) contains as a direct summand a chain complex consisting of a single copy of ℤG in degree m(n + 1), and hence that

H_{G}^{∗}(\underline{E}G, \underline{E}G^{sing}; ℤG) = H^{∗}Hom_{G}(C_{∗}(X^{m}, (X^{m})^{sing}), ℤG)

is non-zero in degree m.(n + 1), and so \underline{cd}G ≥ m.(n + 1).

The inequalities proved so far, together with the inequalities vcdG ≤ F-cdG ≤ \underline{cd}G ≤ \underline{gd}G (which hold for any virtually torsion-free group [9]), imply the claimed equations.

Remark. As remarked earlier, [9] exhibits groups G for which

vcdG = F-cdG = \underline{cd}G = 2, \underline{gd}G = 3.

Those examples were based around the fact that no algebraic notion of dimension is able to capture the distinction between acyclic spaces and contractible spaces, an Eilenberg-Ganea type phenomenon. In contrast, examples coming from Theorem 6 show that vcdG is not the best algebraic approximation to \underline{gd}G.

7. EXAMPLES

Throughout this section, A_{n} and Σ_{n} will denote the alternating and symmetric groups of degree n respectively, Fr_{n} will denote the free group of rank n, and C_{n} will denote the cyclic group of order n.

Example 1. Our motivating example was the following non-simplicial action. Let Q = Σ_{n}, let L be the join of n pairs of points (an n-dimensional analogue of the octahedron), and let Q act by permuting the n pairs. In this case G_{L} is a direct product of n copies of Fr_{2}, and G̃ is the wreath product Fr_{2} ≀ Σ_{n}. The fact that for this L, H_{L} is of type F_{n−1} but not of type F_{n} was known long before the work of Bestvina and Brady. In the case n ≤ 3 it is due to J. Stallings and for general n to R. Bieri [54,7]. In this case, the centralizer C of Σ_{n} in G̃ is the diagonal copy of Fr_{2}, and ϕ_{L}(C) is the index n subgroup of ϕ_{L}(G̃) ≅ ℤ. It follows that C ∩ H_{L} is free of infinite rank, and hence that the property F_{m} is not inherited by centralizers of finite subgroups.

Example 2. One well-known example of a virtually torsion-free group containing infinitely many conjugacy classes of finite subgroups is the infinite free product C_{n} ∗ C_{n} ∗ C_{n} ∗ ⋯, for n > 1. The kernel of the map to C_{n} which is the identity map on each factor is isomorphic to Fr_{∞}. This group fits into the framework of Theorem 3 as follows. Take L to be n points, and let Q = C_{n} permute these points freely. Then G_{L} is isomorphic to Fr_{n}, and H_{L} is isomorphic to Fr_{∞}. The split extension G̃ = G_{L} ⋊ C_{n} is isomorphic to the free product ℤ ∗ C_{n}, and ϕ : ℤ ∗ C_{n} → ℤ is the homomorphism defined by the identity map on ℤ and the trivial map on C_{n}. Hence H̃ is isomorphic to an infinite free product of copies of C_{n}, as claimed.

Before describing our main examples, we recall some results concerning actions of finite groups on contractible complexes. Let 𝔽_{p} denote the field of p elements. P. A. Smith showed that the fixed point set for any action of a p-group on any finite-dimensional 𝔽_{p}-acyclic complex must be 𝔽_{p}-acyclic [53]. Conversely, L. E. Jones showed that any finite 𝔽_{p}-acyclic complex arises as the fixed-point set for some action of C_{p} on a finite contractible complex [30]. In [46] R. Oliver showed that for any finite group Q that is not of prime-power order, there exists an integer n_{Q} ≥ 0 such that a finite complex M is equal to the fixed point set for some action of Q on a finite contractible complex if and only if the Euler characteristic of M satisfies χ(M) ≡ 1 modulo n_{Q}. In [46], the groups for which n_{Q} = 0 and n_{Q} = 1 were determined, together with a characterization of the primes that divide n_{Q} for any Q. In a later paper [47], Oliver was able to compute n_{Q} for all Q.

Example 3. The finite groups Q for which Oliver’s integer n_{Q} is 1 are precisely the groups that are not of the form (p-group)-by-cyclic-by-(q-group), for primes p and q. The smallest such group is A_{5}, and the smallest such abelian group is C_{30} × C_{30}. Any Q for which n_{Q} = 1 acts on some finite contractible complex L without a fixed point, and acts on some finite contractible complex L_{M} with any specified fixed point complex M. (Remark: the second statement may also be deduced directly from the first: take L_{M} to be the join of L and M, with the trivial Q-action on M.)

If Q is any finite group for which n_{Q} = 1, by suitable choice of L we obtain groups H̃ = H_{L} ⋊ Q where H_{L} is of type F, but any one of the following holds:

(a) H̃ contains infinitely many conjugacy classes of subgroup isomorphic to Q;

(b) for any n ≥ 0, the centralizer in H̃ of Q is type F_{n} but not F_{n+1};

(c) the centralizer of Q has any combination of homological finiteness properties that are enjoyed by some Bestvina-Brady group H_{M}. These examples give negative answers to Problems 7.2, 7.4, 7.5 of [38], which concerned the question

of whether the property of having a finite (or finite type) model for \underline{E}G passes to finite extensions. They also give a negative answer to the case ‘finite subgroups’ of problem F7 of [58], which asked whether finite subgroups of a group of type VFP over ℤ or FP over ℚ fall into finitely many conjugacy classes. A related question was recently posed in [10] by M. Bridson: is there, for some n, a subgroup of type F_{∞} of SL_{n}(ℤ) containing infinitely many conjugacy classes of finite subgroups? In Corollary 8 it will be shown that for any L and Q, the group H̃ embeds in SL_{n}(ℤ) for n sufficiently large. From this it follows that these groups afford a positive answer to Bridson’s question.

Note also that for groups H̃ for which L is contractible but L^{Q} is not, the level set X_{t} is a finite proper H̃-CW-complex that is contractible, but there is no finite (or finite type) model for \underline{E}H̃. Hence these groups give a negative answer to two of the three cases of problem 7.8 of [38]. This problem is also implicit in A. Adem’s paper [1]. The main theorems of [1] are proved for groups G such that there is a finite \underline{E}G, although the (now seen to be weaker) hypothesis that there exists some finite proper contractible G-CW-complex is also mentioned.

Example 4. Suppose now that Q is a finite group that is not of prime power order, but that Oliver’s n_{Q} ≠ 1. The smallest such groups are C_{6} and Σ_{3}. Such a Q cannot act without a global fixed point on any finite contractible complex L, but for any M with χ(M) = 1 there is an action of Q on such an L with L^{Q} = M. For any n ≥ 0, there is a finite complex M with χ(M) = 1 such that M is (n − 1)-connected but not n-connected. (Eg., for n = 0 take M = S^{0} ∨ S^{1}, and for n ≥ 1 take M = Σ^{n−1}ℝP^{2}.)

Thus by a suitable choice of L we obtain groups H̃ = H_{L} ⋊ Q of type VF in which for any n ≥ 0, the centralizer C_{H}(Q) is type F_{n} but not type F_{n+1}. Note however, that for these groups Q our methods cannot construct a group of type VF containing infinitely many conjugacy classes of subgroup isomorphic to Q.

These groups also provide a negative answer to problems 7.2, 7.5 and some cases of 7.8 of [38]. In the case when Q is cyclic they provide a negative answer to problem E11 of [58], which asked if the centralizers of elements of finite order in a group of type VFP over ℤ are necessarily of type VFP.

Example 5. Suppose now that Q = C_{p} for p a prime. For any n ≥ 1, there is a finite 𝔽_{p}-acyclic complex which is (n − 1)-connected but not n-connected. The result of L. E. Jones mentioned above implies that for any n ≥ 1 there is a choice of L for which H_{L} is of type F, but the centralizer in H̃ = H_{L} ⋊ C_{p} of an element of order p is type F_{n} but not type F_{n+1}. The difference between these examples and those discussed in Example 4 is that here we require n ≥ 1. In other words, we are unable to find an element of prime power order in a group of type VF whose centralizer is not finitely generated.

Example 6. Examples 3–5 show that the property of having a finite type model for \underline{E}G is not inherited by finite extensions. There is another interesting question concerning extensions. In [36], W. Lück showed that for any extension N ↣ G ↠ \overline{G} of discrete groups, if N and G are both of type F_{∞}, then \overline{G} is also of type F_{∞}. Problem 7.7 of [38] asks if the same property holds for \underline{E}(−). The examples given above in 3–5 show that there exist pairs H_{L}, Q such that H_{L} is of type F, Q is finite, and there is no finite type \underline{E}(H_{L} ⋊ Q). Suppose we are given such a pair H_{L}, Q. Now let N ↣ Fr ↠ Q be a presentation for Q, where Fr is a finitely generated free group (and hence so is N). Let G be the split extension H_{L} ⋊Fr, where Fr acts via the action of Q = Fr/N. Since H_{L} and Fr are both of type F, so is G. The standard copy of N in G centralizes H_{L}, so is normal in G, and \overline{G} = G/N ≅ H_{L} ⋊ Q. Here there are finite models for \underline{E}N and \underline{E}G, but no model for \underline{E}\overline{G} of finite type.

Example 7. In lemma IX.13.2 of [15], K. S. Brown showed that any group G containing a finite-index torsion-free normal subgroup H such that H_{∗}(H; 𝔽_{p}) is finite can contain only finitely many conjugacy classes of p-subgroups. This homological hypothesis on H is considerably weaker than insisting that H be of type FP over 𝔽_{p}. (See also the appendix to [27], where certain homological conditions are shown to imply that a group contains only finitely many conjugacy classes of elementary abelian p-subgroups.)

From the point of view of our examples, the case when H_{L} is FP over 𝔽_{p} corresponds to the case in which L is 𝔽_{p}-acyclic. P. A. Smith’s theorem implies the following, for any finite Q. If the group H_{L} is FP over 𝔽_{p}, then the group H_{L} ⋊ Q contains only finitely many conjugacy classes of p-subgroups, and the centralizer of any p-subgroup is VFP over 𝔽_{p}. This confirms a special case of K. S. Brown’s result, and also suggests: Question 1. If G is any group of type VFP over 𝔽_{p}, and P is a p-subgroup of G, is the centralizer C_{G}(P) necessarily VFP over 𝔽_{p}?

Example 8. Consider the case when Q = C_{p} for p a prime, acting freely on some flag triangulation L of the n-sphere (here n must be odd if p ≠ 2). For each n, this gives rise to a group H̃ = H_{L} ⋊ C_{p} which is of type F_{n} and contains infinitely many conjugacy classes of elements of order p. (Contrast this with K. S. Brown’s result mentioned in Example 7.)

Example 9. The case when H̃ is of type FP over ℚ corresponds to the case when L is ℚ-acyclic. Since L is finite, L is ℚ-acyclic if and only if L is 𝔽_{p}-acyclic for some prime p, if and only if L is 𝔽_{p}-acyclic for almost all primes p. From [46], any finite M with χ(M) = 1 can occur as the fixed-point set for an action of C_{p} on some 𝔽_{q}-acyclic L, for any distinct primes p and q. In particular, the fixed point sets need not be connected. Hence our construction gives rise to groups of type FP over ℚ in which the centralizers of some elements of prime order are not finitely generated. (Contrast this with Example 5.)

Example 10. In (7.6) of [4] (see also problem E1 of [58], and [3]), H. Bass conjectured that any group of type FP over ℚ contains only finitely many conjugacy classes of elements of finite order. Our methods cannot produce a counterexample to this conjecture, since the fixed point set for any action of C_{n} on a finite ℚ-acyclic complex L satisfies χ(L^{C_{n}}) = 1 (see lemma 1 of [46]). Instead, we suggest a strengthening of the conjecture.

Let k be a field of characteristic zero, let G be a (discrete) group, and let Φ = Φ(G) be the Frobenius category of G, i.e., the category whose objects are the finite subgroups of G and whose morphisms are the group homomorphisms induced by conjugation and inclusion in G. Let Φ′ be the full subcategory of Φ whose objects are the finite cyclic subgroups of G. For each such k and G, there is a sequence of homomorphisms as follows:

lim_{C∈Φ′} K_{0}(kC) ⟶^{i} lim_{P ∈Φ} K_{0}(kP) ⟶^{ι_{G}} K_{0}(kG).  (1)

The Artin induction theorem for kP -modules ([52], 9.2 and 12.5), implies that the cokernel of i is a torsion group. It is a well-known conjecture that ι_{G} is always an isomorphism (see for example [37], where this conjecture is described as a case of the Farrell-Jones conjectures).

In the case when G contains only finitely many conjugacy classes of elements of finite order, the category Φ′(G) is skeletally finite, and so the left-hand term in (1) is a finitely generated abelian group, and the middle term is an abelian group of finite rank. In fact, the rank can be described as the number of ‘k-conjugacy classes’ of elements of finite order in G. For the definition of k-conjugacy classes, and the case of a finite group, see [52] 12.4. For a proof of this claim, see [37] section 8, or [1] which considers only the case when k = ℂ, and uses stronger hypotheses. Hattori-Stallings traces can be used to show that the image of ι_{G} has this same rank (a similar argument, although expressed in terms of G_{0} instead of K_{0}, also occurs in [12]).

Now suppose that Q is a finite group that admits a simplicial action without a global fixed point on some finite ℚ-acyclic L. In this case, the alternating sum of the modules in the chain complex for L gives an expression for k, the trivial module for kQ, as an alternating sum of non-trivial permutation modules. If M is any kQ-module, then applying − ⊗_{k} M to the chain complex for L gives an expression for M, viewed as an element of K_{0}(kQ), as an alternating sum of modules induced up from proper subgroups. Hence K_{0}(kQ) is generated by the images of the K_{0}(kP), where P < Q.

For any G, let Φ′′ be the full subcategory of Φ(G) with objects the subgroups that admit no action without a global fixed point on any finite ℚ-acyclic L. By the remarks above, the natural map

lim_{P ∈Φ′′} K_{0}(kP) ⟶ lim_{Q∈Φ} K_{0}(kQ)  (2)

is surjective. In each of our examples of groups of type FP over ℚ, the category Φ′′ is skeletally finite, and so the limit groups in (2) are finitely generated. This leads to Question 2. For G of type FP over ℚ, and k a field of characteristic zero, is lim_{Q∈Φ} K_{0}(kQ) necessarily a finitely generated group?

It appears to be unknown in general whether there is a bound on the orders of finite subgroups of a group G of type FP over ℚ. P. H. Kropholler has shown that there is such a bound if G is also assumed to be of type FP_{∞} over ℤ ([32], section 5).

Example 11. One may define finiteness conditions using cohomology relative to all finite subgroups, as in the definition of the dimension F-cdG. Say that a group G is of type FFP if there is an integer n and an exact sequence

0 → C_{n} → C_{n−1} → ⋯ → C_{0} → ℤ → 0

of G-modules having the properties that

(a) each C_{i} is a direct summand of a sum of modules induced up from finite subgroups of G;

(b) for each finite subgroup H ≤ G, there is an H-equivariant splitting of the exact sequence;

(c) each C_{i} is finitely generated. Now suppose that Q is a finite group acting simplicially on contractible L such that L^{Q} is empty, and

let G = H_{L} ⋊ Q. As in Section 6, it follows that the cellular chain complex for any level set X_{t} satisfies (a),

(b) and (c). On the other hand, there is no model for \underline{E}G having finite 0-skeleton. Thus we see that even strong finiteness conditions for cohomology relative to all finite subgroups do not imply finiteness conditions for \underline{E}G.

Example 12. To apply Theorem 6 and deduce that there are groups G for which the minimal dimension of \underline{E}G is strictly larger than vcdG, we need to construct pairs (L, Q) such that

(i) L is acyclic and n-dimensional; (ii) there is an n-simplex σ ∈ L in a free Q-orbit; (iii) every simplex in the boundary of σ has non-trivial stabilizer.

For n = 2, the acyclic A_{5}-complex described in Example 4 of Section 9 is such a complex. Other examples for Q = A_{5} may be made by attaching extra orbits of simplices to this one. Higher dimensional examples can be obtained using a join construction. If (L, Q) is an example where L has dimension n, and Q′ is any non-trivial finite group, then the join L ∗ Q′, with Q × Q′ acting via the join of the given action on L and the multiplication action of Q′ on itself, is an example of dimension n + 1. Furthermore, L ∗ Q′ is contractible even if L was not. The simplest case of this construction is also described in Example 4 of Section 9.

In the case when Q is the projective general linear group P GL_{2}(q) over a finite field of order q, M. Aschbacher and Y. Segev construct 2-complexes L that satisfy (ii) and (iii), and almost satisfy (i), in the sense that their reduced homology groups are finite [2]. If (L_{1}, Q_{1}) and (L_{2}, Q_{2}) are two such examples for which the reduced homology groups of L_{1} and L_{2} have coprime orders, then L = L_{1} ∗ L_{2} with Q = Q_{1} × Q_{2} is a 5-dimensional contractible example satisfying (i)–(iii).

Since there are pairs L and Q satisfying (i)–(iii) for which L is 2-dimensional, and others for which L is 3-dimensional and contractible, it follows that for each m there are groups G of type VFP over ℤ for which vcdG = F-cdG = 2m and \underline{cd}G = \underline{gd}G = 3m, and that there are groups G of type VF for which vcdG = F-cdG = 3m and \underline{cd}G = \underline{gd}G = 4m.

8. PRESENTATIONS AND LINEARITY

For a flag complex L, the associated right-angled Coxeter group W_{L} is the group with generators the vertices of L, subject to the relations that each generator has order two and that the ends of every edge commute with each other. Just as in the case of the right-angled Artin group G_{L}, this construction is functorial in L. There is of course a quotient map G_{L} ↠ W_{L}, but another map (to be described below) is of greater interest to us.

Recall that a flag complex is determined by its 1-skeleton, since its higher simplices are just the complete subgraphs of its 1-skelelton. If L is a flag complex, define a flag complex S(L) as follows. For each vertex v of L, there are two vertices v′, v′′ (or v^{(1)}, v^{(2)}) of S(L). There is never an edge between v′ and v′′. If v ≠ w and 1 ≤ i, j ≤ 2, then there is an edge in S(L) joining v^{(i)} and w^{(j)} if and only if there is an edge in L joining v and w.

For example, if L is the 2-simplex, S(L) is the octahedron. In general, each n-simplex of L gives rise to 2^{n+1} distinct n-simplices in S(L). The construction of S(L) is functorial in L, and in particular an action of a group Q on L gives rise to an action of Q on S(L) defined by q · v^{(i)} = (q · v)^{(i)}.

Theorem 7. Let L be a finite flag complex with an effective action of Q. The map defined on generators of G_{L} by v ↦ v′v′′ gives an embedding of G_{L} ⋊ Q into W_{S(L)} ⋊ Q.

Proof. The fact that this defines an embedding of G_{L} in W_{S(L)} is proved by Hsu and Wise in [29]. This embedding is equivariant for the action of Q, and so extends to an embedding of G_{L} ⋊ Q in W_{S(L)} ⋊ Q as claimed.

Corollary 8. Suppose L and Q are as in Theorem 7, and that L has n vertices. Then G_{L} ⋊ Q embeds in the integral special linear group SL_{2n}(ℤ).

Proof. For any flag complex M with m vertices, an embedding of W_{M} in SL_{m}(ℤ) is described in [17], p. 55. Under this embedding, there is a natural correspondence between the standard ℤ-basis for ℤ^{m} and the generators for W_{M}. If P is a group of automorphisms of M, then this embedding extends to a homomorphism ρ : W_{M} ⋊ P → GL_{m}(ℤ) by sending elements of the ‘standard copy of P ’ to permutation matrices. In the case when M = S(L) and P = Q, the image ρ(W_{S(L)} ⋊ Q) is contained in SL_{2n}(ℤ) because the Q-set S(L)^{0} is isomorphic to a disjoint union of two copies of L^{0}, and so Q acts on S(L)^{0} via even permutations. Finally, the composite of ρ and the inclusion of G_{L} ⋊ Q in W_{S(L)} ⋊ Q is an embedding, because ρ(G_{L}) ≅ G_{L}, ρ(Q) ≅ Q, and the intersection of ρ(G_{L}) and ρ(Q) is trivial because G_{L} is torsion-free and Q is finite.

Next we recall from [21] a theorem giving a presentation for H_{L} whenever L is connected, and a finite presentation whenever L is 1-connected. The 1-connected case also appears in [28], and the general case is implicit in the work of Bestvina and Brady [6]. (Note that the proof of Theorem 9 in [21] is independent of [6].)

Theorem 9. Let L be a finite connected flag complex. There is a presentation for H_{L} with one generator for each directed edge e of L subject to the following relations.

1. If ē is the same edge as e with the opposite orientation, then ē = e^{−1};

2. For every directed triangle (e, f, g), the relations efg = gfe = 1;

3. For every l ≥ 4, every directed cycle (e_{1}, …, e_{l}) in L and every n ≠ 0, the relation e_{1}^{n}e_{2}^{n} ⋯ e_{l}^{n} = 1. If L is 1-connected the relations of type 3 are not required. In general, if c_{1}, …, c_{r} are cycles such that

L can be made 1-connected by coning of these cycles, then one only need take the type 3 relations for these cycles (and for all n ≠ 0).

In terms of the generators original generators for G_{L}, the directed edge e corresponds to the element vw^{−1}, where v is the initial vertex of e and w is the terminal vertex of e.

As an example, if L is the boundary of an n-gon for n ≥ 4, then H_{L} has the following infinite presentation:

H_{L} = ⟨ a_{1}, …, a_{n} : a_{1}^{m} ⋯ a_{n}^{m} = 1, for all m ≠ 0 ⟩.

A very similar presentation for this group in the case when n = 4 is given in [54].

If Q acts as a group of automorphisms of L, then Q permutes the generators and relators in the presentations for H_{L} given in Theorem 9. This gives an easy way to produce presentations for the groups H_{L} ⋊ Q: take a ‘fundamental domain’ M for the action of Q (i.e., a subcomplex whose images under Q form the whole of L), and eliminate all the edges and type 2 relations except those coming from M. One example of this is the following.

Corollary 10. Let L be a finite 1-connected complex, with an effective action of Q, and let M be any 1-connected subcomplex of L whose images under Q cover L. A presentation for the group H_{L} ⋊ Q is given by: take the presentation of H_{M} from Theorem 9, use this to make a presentation for the free product H_{M} ∗Q, and for each edge e ∈ M and each q ∈ Q such that the edge e′ = q ·e also lies in M, add the relation qeq^{−1} = e′.

In the examples in the next section, M can be chosen to be an n-simplex for some n, and so H_{L} ⋊ Q will be presented as a quotient of the free product ℤ^{n} ∗ Q.

9. MORE EXAMPLES

In this section, we adopt the same notation for various specific groups as in Section 7. The examples discussed here are special cases of the ones mentioned already in Section 7, although now we are interested in providing explicit triangulations with either very few vertices or very few Q-orbits of simplices.

Example 1. Q = C_{n} for some n > 3, acting freely on L, the boundary of an n-gon. In this case, H̃ = H_{L}⋊Q is a finitely generated (but not finitely presented) group containing, for each m > 1 that divides n, infinitely many conjugacy classes of elements of order m. Moreover, H̃ embeds in SL_{2n}(ℤ). Let x be a generator for C_{n}, and suppose that (using the presentation for H_{L} given at the end of the previous section) x^{−1}a_{i}x = a_{i+1} for each i, where indices should be taken modulo n. Eliminating a_{2}, …, a_{n} gives the following presentation for H̃.

H̃ = ⟨ a, x : x^{n} = 1, (a^{m}x^{−1})^{n−1}a^{m}x^{n−1} = 1, for all m ≠ 0 ⟩.

This can be simplified to give

H̃ = ⟨ a, x : (a^{m}x^{−1})^{n} = 1, for all m ∈ ℤ ⟩.

One can also find explicit words for elements of given order that lie in distinct conjugacy classes. Write v_{1}, …, v_{n} for the generators of G_{L}, so that a_{i} = v_{i−1}v_{i}^{−1} and x^{−1}v_{i}x = v_{i+1} for all i, where as before indices are taken mod n. Since L^{Q} is empty, each element of G_{L} ⋊ Q fixes a single point of X_{L}. A study of the proof of Theorem 3 shows that the elements v_{1}^{−l}xv_{1}^{l} for l ∈ ℤ fix points that are at different heights, and so these elements generate distinct conjugacy classes of cyclic subgroups of H̃. It remains to express these elements in terms of the generators for H̃, but we have that xv_{1}x^{−1} = v_{n}, and so xv_{1}^{l} = v_{n}^{l}x. Hence

v_{1}^{−l}xv_{1}^{l} = v_{1}^{−l}v_{n}^{l}x = a^{−l}x.

Thus we see that the elements { a^{l}x : l ∈ ℤ } generate distinct conjugacy classes of subgroups of H̃ of order n, and that any subgroup of H̃ of order n is conjugate to one of these.

Example 2. Actions of Q = C_{2} on the (n − 1)-sphere L, triangulated as a join of n pairs of points. Here H_{L} is the subgroup of (Fr_{2})^{n} that is F_{n−1} but not F_{n} that was introduced by Stallings for n ≤ 3 and Bieri for general n [54,7]. For any m ≤ n, there is a simplicial action of C_{2} on L which swaps the members of m of the pairs of points, and fixes the members of the other pairs. These actions give rise to index two extensions of H_{L} containing one of:

(a) infinitely many conjugacy classes of elements of order two (in the case m = n);

(b) elements of order two whose centralizers are F_{n−m−1} but not F_{n−m} (in the case m < n).

Corollary 8 implies that each of these groups embeds in GL_{4n}(ℤ), but this can be improved. In the case n = 1, the group G_{L} ⋊ C_{2} is Fr_{2} ⋊ C_{2} ≅ ℤ ∗ C_{2}, and for general n, these groups are subgroups of (ℤ ∗ C_{2})^{n}. The group ℤ ∗ C_{2} embeds in GL_{2}(ℤ) via the map that sends the generators of the factors to the matrices

( \begin{matrix}1 & 2 \\ 0 & 1 \end{matrix} )  and  ( \begin{matrix}0 & 1 \\ 1 & 0 \end{matrix} ) .

Hence it follows that each of these groups embeds in GL_{2n}(ℤ). In particular, the antipodal action of C_{2} on the square gives rise to a finitely generated subgroup of GL_{4}(ℤ) containing infinitely many conjugacy classes of elements of order two, and the antipodal action of C_{2} on the octahedron gives rise to a finitely presented subgroup of GL_{6}(ℤ) that contains infinitely many conjugacy classes of elements of order two. In [25], F. Grunewald and V. Platonov constructed a finitely generated subgroup of SL_{4}(ℤ) containing infinitely many conjugacy classes of elements of order 4, and asked if there could be a finitely presented subgroup of GL_{n}(ℤ) for any n having this property (see also problem 4 of [48]. Using the example of Grunewald and Platonov, M. Bridson has constructed a finitely presented subgroup of SL_{6}(ℤ) containing infinitely many conjugacy classes of elements of order 4 [10]. The examples of Grunewald-Platonov and Bridson contain only finitely many conjugacy classes of elements of order two.

We consider presentations only in the case n = 3. As remarked earlier, the presentations coming from Theorem 9 are simplest when there are very few Q-orbits of simplices in L. Hence we consider the case of (C_{2})^{3} acting on the octahedron L by reflections in three orthogonal planes that each contain four of its vertices. For this action, the 2-simplices of the octahedron form a single orbit, and the edges form three distinct orbits. Let a and b be edges of a 2-simplex of the octahedron, with the terminal vertex of a and the initial vertex of b equal (so that in H_{L} the third edge of this 2-simplex is equal to ab). Let x, y, and z be reflections in the planes containing a, b and ab respectively. The following describes a presentation for H̃ = H_{L} ⋊ (C_{2})^{3}. Take the free product of ℤ^{2} generated by a and b with (C_{2})^{3} generated by x, y and z, and add the relations:

[a, x] = [b, y] = [ab, z] = 1.

This group H̃ is an index 8 extension of Stallings’ group that is finitely presented but not F_{3}. The centralizer of the element x is finitely generated but not finitely presented, and the centralizer of the element xy is not finitely generated. There are infinitely many conjugacy classes of elements of elements of order two in H̃ that map to xyz in the quotient of H̃ by the normal subgroup generated by a and b (which is isomorphic to (C_{2})^{3}).

There is an explicit embedding of this group H̃ into (GL_{2}(ℤ))^{3} ≤ GL_{6}(ℤ) given by

a ↦ ( \begin{matrix}1 & 2 & · & · & · & · \\ 0 & 1 & · & · & · & · \\ · & · & 1 & −2 & · & · \\ · & · & 0 & 1 & · & · \\ · & · & · & · & 1 & 0 \\ · & · & · & · & 0 & 1 \end{matrix} ) b ↦ ( \begin{matrix}1 & 0 & · & · & · & · \\ 0 & 1 & · & · & · & · \\ · & · & 1 & 2 & · & · \\ · & · & 0 & 1 & · & · \\ · & · & · & · & 1 & −2 \\ · & · & · & · & 0 & 1 \end{matrix} ) ,

and sending x, y and z to the permutation matrices corresponding to the transpositions (5, 6), (1, 2) and (3, 4) respectively.

Finally, for m ∈ ℤ, the elements a^{m}ya^{−m}xz have order two and lie in distinct conjugacy classes in H̃, although they are conjugate in G̃ = G_{L} ⋊ (C_{2})^{3}. This follows from an explicit calculation, as in Example 1, but it is simpler to apply the general statement given below as Lemma 11. Even checking directly that this element has order two is not trivial, but one can check that

(a^{m}ya^{−m}xz)^{2} = a^{m}ya^{−m}za^{m}ya^{−m}z = a^{m}(ya^{−m}y)(yza^{m}zy)(za^{−m}z) = 1,

where the final equality follows because the word as bracketed is equal to a ‘type 3 relation’ in the presentation for H_{L}.

Lemma 11. Let L and Q be as in Theorem 7, suppose that L is connected, and take the presentation for H_{L} as described in Theorem 9. If q ∈ Q and v is one of the vertices of L viewed as an element of G_{L}, then the element v^{m}qv^{−m} can be expressed in terms of the generators for H_{L} ⋊Q as e_{1}^{m} ⋯ e_{l}^{m}q, where (e_{1}, …, e_{l}) is a directed path from v to the vertex w = q · v = qvq^{−1}.

Proof. v^{m}qv^{−m} = v^{m}qv^{−m}q^{−1}q = v^{m}w^{−m}q = (e_{1}^{m} ⋯ e_{l}^{m})q.

Example 3. The group C_{n} for n ≥ 4 acting freely on S^{2m−1}. Here it is convenient to triangulate S^{2m−1} as a join of m copies of the n-gon (as in Example 1). In this way one obtains a subgroup of SL_{2mn}(ℤ) that is type F_{2m−1} (but not type F_{2m}) and contains infinitely many conjugacy classes of elements of order l for each l > 1 dividing n. We give a presentation only in the case m = 2. To minimize the number of relators, it is more convenient to let C_{n} × C_{n} act on L = S^{3}. Let a (resp. c) be an edge of an n-gon, and let x (resp.

y) be a generator for a copy of C_{n} that rotates this n-gon and sends the initial vertex of a (resp. c) to the terminal vertex. Suppose that x fixes c and that y fixes a. Finally, let b be the edge in the join of the two n-gons that goes from the terminal vertex of a to the initial vertex of c. The 3-simplex containing the edges a, b and c is a fundamental domain for the action of C_{n} × C_{n} on L. The group H̃ = H_{L} ⋊ (C_{n})^{2} has the following presentation. Take the free product of a copy of ℤ^{3}, generated by a, b and c, and (C_{n})^{2} generated by x and y. Add the relations

[c, x] = [a, y] = 1,  x^{−1}bx = ab,  yby^{−1} = bc.

The reason why these relations are slightly more complicated than in Example 2 is that four of the edges of the fundamental domain lie in the same orbit. The edges are a, b, c, ab, bc and abc. The action of y sends b to bc and the action of x^{−1} sends b to ab. (Also x^{−1}y sends b to abc, but the corresponding relation can be eliminated from the presentation.) The subgroup of H̃ generated by a and x is the group described in Example 1, and

C_{H}(y) = ⟨a, x⟩ × ⟨y⟩.

The centralizer of x^{−1}y, on the other hand is the copy of (C_{n})^{2} generated by x and y. By Lemma 11, the elements c^{l}x^{−1}y for l ∈ ℤ all lie in distinct conjugacy classes.

Example 4. A fixed-point free action on a contractible complex. There is a well-known 2-dimensional acyclic simplicial complex M with an action of A_{5} without a global fixed point, which will be our starting point. M is described as the barycentric subdivision of a polyhedral complex. Recall that there are two conjugacy classes of 12 elements of order 5 in A_{5}, and that each element x of order 5 is conjugate to x^{−1} (but not to x^{2} or x^{3}), so that each conjugacy class consists of 6 pairs of inverse elements. Take the 1skeleton of a simplex with vertex set {1, …, 5}, and attach 6 pentagonal 2-cells along the unoriented cycles corresponding to one of the two conjugacy classes of elements of order five. By construction, this complex admits an action of A_{5} without a global fixed point. It may be checked that the complex is acyclic and that its fundamental group is isomorphic to SL_{2}(𝔽_{5}), the perfect group of order 120. (See also [55], p34 and [9], sec. 4.) Its barycentric subdivision, M, has 21 0-simplices in 3 A_{5}-orbits, 80 1-simplices in 3 A_{5}-orbits and 60 2-simplices in a single, free A_{5}-orbit. As in Example 12 of Section 7, it follows that the group H̃ = H_{M} ⋊ A_{5} is a group of type VFP with vcdH̃ = 2 but such that any model for \underline{E}H̃ has dimension at least three.

The join of M with any non-empty simplicial complex is contractible. For example, if we take L to be the join of M with a set of 5 points permuted transitively by A_{5}, then L is contractible, has 26 vertices, and L^{A_{5}} = ∅. Hence H̃ = H_{L} ⋊ A_{5} is a subgroup of SL_{52}(ℤ) of type VF containing infinitely many conjugacy classes of subgroup isomorphic to A_{5}. On the other hand, if we take L to be the join of M and two points (with trivial A_{5} action), then H̃ = H_{L} ⋊ A_{5} is a subgroup of SL_{46}(ℤ) of type VF in which the centralizer of an A_{5} is not finitely generated (and is in fact free of infinite rank).

From the point of view of finding presentations, the simplest example seems to be to take Q = A_{5} × C_{2}, and to take L to be the join of M (with Q acting via the A_{5} action) and two points (permuted freely by C_{2}). To be completely explicit, choose the conjugacy class of 5-cycles in A_{5} containing (1, 2, 3, 4, 5) to define M, and take the 2-simplex with vertices (1), (1, 2) and (1, 2, 3, 4, 5) as the fundamental domain for the A_{5}-action on M. Write N and S for the two extra vertices permuted freely by C_{2}, let τ denote the generator of C_{2}, and write elements of A_{5} in cycle notation. In L, let a denote the edge from N to (1, 2, 3, 4, 5), let b denote the edge from (1, 2, 3, 4, 5) to (1, 2), and let c denote the edge from (1, 2) to (1). Then the group H̃ = H_{L} ⋊ (A_{5} × C_{2}) has the following presentation. Take the free product of ℤ^{3} generated by a, b and c with A_{5} × C_{2} and add the following twelve commutators as relators.

[a, (1, 2, 3, 4, 5)]  [a, (1, 2)(3, 5)]  [b, τ]  [b, (1, 2)(3, 5)]
[c, τ]  [c, (3, 4, 5)]  [ab, (1, 2)(3, 4)]  [ab, (3, 4, 5)]
[bc, τ]  [bc, (2, 5)(3, 4)]  [abc, (2, 3)(4, 5)]  [abc, (2, 3, 4)]

This gives an explicit presentation of a group H̃ = H_{L} ⋊ Q of type VF that contains infinitely many conjugacy classes of subgroup isomorphic to A_{5} × C_{2}, and which can be embedded in SL_{46}(ℤ). The group H̃ is of virtual cohomological dimension three, and any model for \underline{E}H̃ has dimension at least four. The centralizer of the element τ is VFP over ℤ, but is not finitely presented. Writing A for the standard copy of A_{5} in H̃ = H_{L} ⋊ Q, the conjugates (in G_{L} ⋊ Q) of A × ⟨τ⟩ by powers of N are the groups A × ⟨a^{m}τa^{−m}⟩. Hence these groups, for m ∈ ℤ, are representatives of the distinct conjugacy classes of A_{5} × C_{2} in H_{L} ⋊ Q. The centralizer C_{H}(A) can also be described in terms of these elements. It is the group H_{{N,S}}⋊⟨τ⟩, which is isomorphic to an infinite free product of copies of C_{2} generated (freely) by the elements a^{m}τa^{−m} for m ∈ ℤ.

10. K-THEORY

In this section, we construct groups G of type VF for which the rational homology groups of centralizers of certain elements of finite order are not finite-dimensional. We deduce that these groups have the properties that their higher algebraic K-groups K_{i}(RG) for various rings R and the (topological) K-groups of their reduced group C^{∗} algebras K_{∗}^{Top}(C_{r}^{∗}(G)) are not of finite rank.

Proposition 12. Let L be a finite flag complex, and Q a group of automorphisms of L. The Baum-Connes conjecture (with coefficients) holds for the groups H_{L} ⋊ Q and G_{L} ⋊ Q.

Proof. N. Higson and G. Kasparov have shown that the Baum-Connes conjecture holds for any discrete group admitting a properly discontinuous action on a Hilbert space by affine isometries [26,31]. (Groups admitting such an action are said to be ‘a-T-menable’ or to ‘have the Haagerup property’.) An argument due to G. Niblo and L. Reeves shows that any group admitting a proper action on a CAT(0) cube complex has the Haagerup property (see the ‘Technical Lemma’ in [44], where the Haagerup property is not explicitly mentioned, or section 1.4 of [31]). Since G_{L} ⋊ Q acts properly on the cube complex X_{L}, it follows that the Baum-Connes conjecture holds for G_{L} ⋊ Q and for each of its subgroups.

Proposition 13. Let L be a flag complex homeomorphic to the (n − 1)-sphere for some n > 0. Then the rational homology group H_{n}(H_{L}; ℚ) is infinite dimensional.

Proof. The argument is based on the one used by Stallings in the case when L is the octahedron [54], see also corollary 6 of [21]. Throughout this proof, homology is to be understood to be taken with rational coefficients, which will be omitted. The complex X_{L} is an n-dimensional model for EH, and by the Bestvina-Brady theorem (quoted as Theorem 1), H_{L} is of type F_{n−1}. Hence the homology groups H_{i}(H_{L}) are finitedimensional for i < n and are trivial for i > n.

Now suppose that H_{n}(H_{L}) is also finite-dimensional. In this case, H_{L} has a well-defined Euler characteristic χ(H_{L}). The expression for G_{L} as an extension with kernel H_{L} and quotient ℤ gives rise to a long exact sequence

⋯ → H_{i}(H_{L}) → H_{i}(H_{L}) → H_{i}(G_{L}) → H_{i−1}(H_{L}) → H_{i−1}(H_{L}) → ⋯,

and so the Euler characteristic χ(G_{L}) must be zero. However, a count of the cells in X_{L}/G_{L} shows that χ(G_{L}) = 1 − χ(L) = (−1)^{n}. This contradiction shows that H_{n}(H_{L}) cannot be finite-dimensional.

The formula χ(G_{L}) = 1 − χ(L), which holds for any L, predates the work of Bestvina and Brady. It was proved by C. Droms in [22].

Proposition 14. Suppose that the flag complex L is the disjoint union of non-empty subcomplexes M and N. Then H_{L} is isomorphic to the free product of H_{M}, H_{N} and a free group of infinite rank. In symbols, H_{L} ≅ H_{M} ∗ H_{N} ∗ Fr_{∞}.

Proof. The group G_{L} is isomorphic to the free product G_{M} ∗ G_{N}. Let Y and Z be models for K(G_{M}, 1) and K(G_{N}, 1) respectively, and as a model for K(G_{L}, 1) take X consisting of Y and Z joined by an interval I. Consider the infinite cyclic cover X̃ of X corresponding to the subgroup H_{L}. Since ϕ_{L} restricted to G_{M} (resp. to G_{N}) is equal to ϕ_{M} (resp. ϕ_{N}), the part of X̃ lying above Y (resp. Z) is a model for K(H_{M}, 1) (resp. K(H_{N}, 1)). Thus X̃ consists of a copy of K(H_{M}, 1) and a copy of K(H_{N}, 1) joined together by infinitely many intervals. The fundamental group of this space is as claimed.

Corollary 15. For each n ≥ 1, there is a flag complex M with the properties that χ(M) = 1 and H_{n}(H_{M}; ℚ) is not finite-dimensional.

Proof. When n is even, take M to be a triangulation of the disjoint union of S^{n−1} and a point. When n is odd, take M to be a triangulation of the disjoint union of S^{n−1} and a bouquet of two circles. The claim concerning the homology of H_{M} follows from Propositions 13 and 14.

Example. For any finite flag complex M with χ(M) = 1, there is an action of the cyclic group C_{6} on a finite contractible flag complex L such that L^{C_{6}} = M (see [46]). In H̃ = H_{L} ⋊ C_{6}, the centralizer of any element of order 6 is isomorphic to H_{M} × C_{6}. For any n > 0, if we choose M as in Corollary 15, we obtain a group H̃ = H̃(n) of type VF and an element g ∈ H̃ of order six such that H_{n}(C_{H}(g); ℚ) is infinite-dimensional.

In theorem 0.1 of [37], W. Lück constructs, for any discrete group G, a homomorphism

⨁_{g∈C} ⨁_{i+j=n} H_{i}(C_{G}(g); ℂ) ⊗ K_{j}^{Top}(ℂ) ⟶ ℂ ⊗ K_{n}^{Top}(C_{r}^{∗}(G)).  (3)

Here K_{∗}^{Top}(C_{r}^{∗}(G)) denotes the topological K-groups of the reduced C^{∗}-algebra for G, the indexing set C is a set of representatives for the conjugacy classes of elements of finite order in G, and C_{G}(g) denotes the centralizer of g. This homomorphism is an isomorphism whenever the Baum-Connes conjecture is valid for G. From Proposition 12, we see that the group H̃ = H̃(n) constructed above has the property that K_{n}^{Top}(C_{r}^{∗}(H̃)) has infinite rank.

One may also apply these techniques to deduce that there are groups of type VF for which the algebraic K-groups K_{∗}(RG) have infinite rank. Of course, this is only of interest in the case when R is such that each K_{i}(R) has finite rank: if R is a field of characteristic zero, and G is free abelian of rank n − 1, then K_{n}(RG) has infinite rank, since it contains K_{1}(R) as a direct summand.

Two theorems of Quillen (see 5.3.12 and 5.3.27 in [49]) together imply that if R is obtained from the ring of integers in any algebraic number field by inverting finitely many primes, then each K_{i}(R) is finitely generated.

M. Matthey has recently proved the following [41,40]. Let m be a positive integer, and let G be a group in which the order of every torsion element divides m. For any subring R of ℂ that contains both 1/m and exp(2πi/m), there is an injective map

⨁_{g∈C} H_{n}(C_{G}(g); ℂ) → ℂ ⊗ K_{n}(RG),  (4)

where the indexing set C is as in (3). In particular, for R = ℤ[1/6, exp(2πi/3)] and H̃ = H̃(n) as above, the group K_{n}(RH̃) is not of finite rank, although each K_{i}(R) is finitely generated. M. Matthey has shown the authors a modified version of (4) that applies in the case R = ℤ[1/m], which implies that K_{n}(ℤ[1/6]H̃) is not of finite rank.

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Authors’ addresses I. J. Leary, Faculty of Mathematical Studies, University of Southampton, Southampton, SO17 1BJ, U.K. B. E. A. Nucinkis, Departement Mathematik, ETH Zürich, CH-8092 Zürich, Switzerland

Email addresses

I.J.Leary@maths.soton.ac.uk    bean@math.ethz.ch