Finite Virtual Cohomological Dimension Research Articles

AUTOMORPHISMS OF PRO-p GROUPS OF FINITE VIRTUAL COHOMOLOGICAL DIMENSION

AbstractLet G be a pro-p group of finite cohomological dimension and type FP ∞ and T be a finite p-group of automorphisms of G. We prove that the group of fixed points of T in G is again a pro-p group of type FP ∞ (in particular it is finitely presented). Moreover, we prove that a pro-p group G of type FP ∞ and finite virtual cohomological dimension has finitely many conjugacy classes of finite subgroups.

On a conjecture of E. Rapaport Strasser about knot-like groups and its pro- p version

A group G is knot-like if it is finitely presented of deficiency 1 and has abelianization G / G ′ ≃ Z . We prove the conjecture of E. Rapaport Strasser that if a knot-like group G has a finitely generated commutator subgroup G ′ then G ′ should be free in the special case when the commutator G ′ is residually finite. It is a corollary of a much more general result : if G is a discrete group of geometric dimension n with a finite K ( G , 1 ) -complex Y of dimension n, Y has Euler characteristics 0, N is a normal residually finite subgroup of G, N is of homological type FP n - 1 and G / N ≃ Z then N is of homological type FP n and hence G / N has finite virtual cohomological dimension vcd ( G / N ) = cd ( G ) - cd ( N ) . In particular either N has finite index in G or cd ( N ) ⩽ cd ( G ) - 1 . Furthermore we show a pro- p version of the above result with the weaker assumption that G / N is a pro- p group of finite rank. Consequently a pro- p version of Rapaport's conjecture holds.

Multiple complexes and gaps in Farrell cohomology

Let Γ be a group of finite virtual cohomological dimension n. In this paper, we show that Γ has a multiple complex, i.e. Γ has a projective resolution that is the tensor product of γ complexes which are resolutions of some period after n-steps, where γ is the maximum of the p-ranks of the finite elementary abelian p-subgroups of Γ. This is an extension of the results of Benson and Carlson (Math. Z. 195 (1987) 221) and Benson and Greenless (J. Algebra 192 (1997) 678). Finally, we present a result on gaps in Farrell cohomology using a multiple complex. This result is a generalization of the result of Benson et al. (J. Algebra 131 (1990) 40) and Juan-Pineda (J. Pure and Appl. Algebra 111 (1996) 213).

Cohomology of discrete groups and their finite subgroups

We investigate the cohomology of a group having finite virtual cohomological dimension in terms of the contributions from finite subgroups. As a result, we prove a variant of Quillen's F-isomorphism theorem which remains valid for an arbitrary commutative ring of coefficients and for suitable families of finite subgroups.

Discreteness properties of translation numbers in solvable groups

We define a group to be translation proper if it carries a leftinvariant metric in which the translation numbers of the non-torsion elements are nonzero and translation discrete if they are bounded away from zero. The main results of this paper are that a translation proper solvable group of finite virtual cohomological dimension is metabelian-by-finite, and that a translation discrete solvable group of finite virtual cohomological dimension, m, is a finite extension of Zm.

Aspherical Coxeter Groups That are Quillen Groups

Let Γ be a group having finite virtual cohomological dimension. Let BΓ be its classifying space, and let C(Γ) be its Quillen category, whose objects are elementary abelian 2-subgroups V of Γ and whose morphisms are restrictions of conjugacy of Γ. The group Γ is called a Quillen group if the natural map [formula] is an isomorphism. The symmetric groups (see [8]) and the Coxeter groups of finite order (see [7]) are examples of Quillen groups. J. H. Gunawardena, J. Lannes and S. Zarati obtained the characterization of Quillen groups in terms of unstable modules over the mod 2 Steenrod algebra [8] (see Proposition 2 below). By making use of this result, M. Errokh and F. Grazzini obtained the classification of Coxeter groups on three generators which are Quillen groups [8]. The purpose of the present paper is to obtain a necessary and sufficient condition for aspherical Coxeter groups to be Quillen groups (Theorem 1). Our result is very simple, and the previously mentioned result by Errokh and Grazzini follows easily from ours (see Example 1). The paper is organized as follows. In Section 2, after giving relevant definitions concerning Coxeter groups, we shall state our main result. The next two sections are devoted to preliminaries. In Section 3, we shall review necessary definitions and facts concerning unstable modules over the mod 2 Steenrod algebra. In Section 4, we shall discuss the Leray spectral sequences converging to the cohomology of Coxeter groups. The proof of Theorem 1 will be given in Section 5. In Section 6, we shall give examples illustrating Theorem 1. NOTATION. The cardinality of a finite set X is denoted by ∣X∣. 1991 Mathematics Subject Classification 20F55.

On p -Completed Classifying Spaces of Discrete Groups and Finite Complexes

It is shown that for certain discrete p-perfect groups G, in particular for all p-perfect finite groups and certain groups of finite virtual cohomological dimension, the loop space on the p-completed classifying space Ω B G p ^ is a retract of the loop space on the p-completion of a certain finite complex. For those finite virtual cohomological dimension groups where the author's methods apply, a bound on the dimension of such a complex is provided.

A note on the homotopy type of ΩBSL 3(ℤ)and 2

It is known that for p-perfect groups G of finite virtual cohomological dimension and finite type mod-p cohomology, the p-completed classifying space BGandp has the property that ΩBGandp is a retract of the loop space on a simply-connected, [ ]p-finite, p-complete space. In this note we consider a particular example where this theorem applies, namely we study the homotopy type of BSL3(ℤ)and2. It particular we analyse ΩBSt3(ℤ)and2, a double cover of ΩBSL3(ℤ)and2, and obtain a splitting theorem for it in terms of 2-primary Moore spaces and fibres of degree 2r maps on spheres. We also give a formula for the Poincaré series of H∗(ΩBΓandp; [ ]p) for a general group Γ, as above, in terms of possibly simpler components. This formula is used to calculate the mod-2 homology of ΩBΓand2 for Γ=SL3(ℤ) or St3(ℤ) as modules over a certain tensor subalgebra.

On the ring structure of [formula omitted] for discrete groups

Let Γ be a discrete group of finite virtual cohomological dimension. Using a rational splitting for G-equivariant K-theory (when G is a finite group) due to T. tom Dieck, we find a rational splitting for the classifying space of such groups.

Homological Finiteness Conditions for Modules Over Group Algebras

We develop a theory of modules of type FP? over group algebras of hierarchically decomposable groups. This class of groups is denoted HF and contains many different kinds of discrete groups including all countable polylinear groups. Amongst various results, we show that if G is an HF-group and M a ZG-module of type FP? then M has finite projective dimension over ZH for all torsion-free subgroups H of G. We also show that if G is an HF-group of type FP? and M is a ZG-module which is ZF-projective for all finite subgroups F of G, then M has finite projective dimension over ZG. Both of these results have as a special case the striking fact that if G is an HF-group of type FP? then the torsion-free subgroups of G have finite cohomological dimension. A further result in this spirit states that every residually finite HF-group of type FP? has finite virtual cohomological dimension.

On the finite generation of high dimensional cohomology ring of virtually torsion-free groups

Let Γ be a discrete group and p be a prime. One of the fundamental results in group cohomology is that H*(Γ, [ ]p) is a finitely generated [ ]p-algebra if Γ is a finite group [8, 24]. The purpose of this paper is to study the analogous question when Γ is no longer finite.Recall that Γ is said to have finite virtual cohomological dimension (vcd) if there exists a finite index torion-free subgroup Γ′ of Γ such that Γ′ has finite cohomological dimension over ℤ [4]. By definition vcd Γ is the cohomological dimension of Γ′. It is easy to see that the mod p cohomology ring of a finite vcd-group does not have to be a finitely generated [ ]p-algebra in general. For instance, if Γ is a countably infinite free product of ℤ's, then H1(Γ, [ ]p) is not finite dimensional over [ ]p. The three most important classes of examples of finite vcd-groups in which the mod p cohomology ring is a finitely generated [ ]p-algebra are arithmetic groups [2], mapping class groups [9, 10] and outer automorphism groups of free groups [5]. In each of these examples, the proof of finite generation involves the construction of a specific Γ-complex with appropriate finiteness conditions. These constructions should be regarded as utilizing the geometry underlying these special classes of groups. In contrast, the result we prove will depend only on the algebraic structure of the group Γ.

A homotopy decomposition for the classifying space of virtually torsion-free groups and applications

Let Γ be a discrete group. Γ is said to have finite virtual cohomological dimension (vcd) if there exists a finite index torsion-free subgroup Γ' of Γ such that Γ' has finite cohomological dimension over. Examples of such groups include the fundamental group of a finite graph of finite groups, arithmetic groups, mapping class groups and outer automorphism groups of free groups. One of the fundamental problems in topology is to understand the cohomology of these finitevcd-groups.

Cohomological nonvanishing for modules over discrete groups

Let Γ be a discrete group of finite virtual cohomological dimension, and let V be a f.g. Z-torsion free Γ module. In this situation, the Farrell cohomology of Γ with coefficients in V is defined and we prove in this paper cohomological non-vanishing results for these groups similar to those existing for finite groups and Tate cohomology, i.e. that we either have that these groups vanish for all dimensions or there are infinitely many nontrivial. The proof is based on a geometric approach to these groups, the study of Euler characteristics, minimal resolutions, and the notion of exponent.

On a conjecture of Moore

This can be regarded as a generalization of Serre’s Theorem that every torsion-free group of finite virtual cohomological dimension has finite cohomological dimension (see [2]). For suppose that G is torsion-free and that H has cohomological dimension n < co. Then the nth syzygy in any projective resolution of Z over ZG is projective as ZH-module, and Moore’s Conjecture implies at once that it is also projective as ZG-module. One of the special cases of the conjecture which we prove here also implies Serre’s Theorem. It is natural to generalize the conjecture, because, in addition to group rings, it makes perfect sense for crossed products and in fact for strongly group-graded rings. Recall that a G-graded ring is a ring A with a direct sum decomposition

Farrell cohomology and centralizets of elementary abelian p-subgroups

Let Γ be a discrete group. Γ is said to have finite virtual cohomological dimension (vcd) if there exists a finite index torsion-free subgroup Γ′ of G such that Γ′ has finite cohomological dimension over ℤ. Examples of such groups include finite groups, fundamental group of a finite graph of finite groups, arithmetic groups, mapping class groups and outer automorphism groups of free groups. One of the fundamental problems in topology is to understand the cohomology of these finite vcd-groups.

Complexity and varieties for infinitely generated modules

In the past fifteen years the theory of complexity and varieties of modules has become a standard tool in the modular representation theory of finite groups. Moreover the techniques have been used in the study of integral representations [8] and have been extended to the representation theories of objects such as groups of finite virtual cohomological dimension [1], infinitesimal subgroups of algebraic groups and restricted Lie algebras [14, 16]. In all cases some sort of finiteness condition on the module category has been required to make the theory work. Usually this comes in the form of stipulating that all modules under consideration be finitely generated. While the restrictions have been efficient for most applications to date, there are very good reasons for wanting to develop a theory that will accommodate infinitely generated modules. One reason might be the possibility of extending the techniques of representations to other classes of infinite groups. Another reason is that some recent work has revealed a few of the defects of the finiteness requirement. One such problem can be summarized as follows.

On the Yagita invariant of mapping class groups

Let Γ denote a group of finite virtual cohomological dimension and p a prime. If the cohomology ring H∗(Γ;Fp) has Krull dimension one, the p-period of Γ is defined; it measures the periodicity of H∗(Γ;Fp) in degrees above the virtual cohomological dimension of Γ. The Yagita invariant p(Γ) of Γ is a natural generalization of the p-period to groups with H∗(Γ;Fp) of Krull dimension larger than one. We compute the Yagita invariant p(Γg) for the mapping class group Γg with respect to an arbitrary odd regular prime p.

Tate cohomology for arbitrary groups via satellites

We define cohomology groups Ĥ n ( G; M), n ϵ Z , for an arbitrary group G and G-module M, using the concept of satellites. These cohomology groups generalize the Farrell-Tate groups for groups of finite virtual cohomological dimension and form a connected sequence of functors, characterized by a natural universal property.

Characters andK-theory of discrete groups

Let $\Gamma$ be a discrete group of finite virtual cohomological dimension with certain finiteness conditions of the type satisfied by arithmetic groups. We define a representation ring for $\Gamma$, determined on its elements of finite order, which is of finite type. Then we determine the contribution of this ring to the topological $K$-theory $K^*(B\Gamma)$, obtaining an exact formula for the difference in terms of the cohomology of the centralizers of elements of finite order in $\Gamma$.

Representations and 𝐾-theory of discrete groups

Let Γ \Gamma be a discrete group of finite virtual cohomological dimension with certain finiteness conditions of the type satisfied by arithmetic groups. We define a representation ring for Γ \Gamma , determined on its elements of finite order, which is of finite type. Then we determine the contribution of this ring to the topological K-theory K ∗ ( B Γ ) K^{\ast }(B\Gamma ) , obtaining an exact formula for the difference in terms of the cohomology of the centralizers of elements of finite order in Γ \Gamma .