Peter H. Kropholler

On groups of type (FP)

Let G be a group. A ZG-module M is said to be of type (FP)? over G if and only if there is a projective resolution P? ?M in which every Pi is finitely generated. We show that if G belongs to a large class of torsion-free groups, which includes torsion-free linear and soluble-by-finite groups, then every ZG-module of type (FP)? has finite projective dimension. We also prove that every soluble or linear group of type (FP)? is virtually of type (FP). The arguments apply to groups which admit hierarchical decompositions. We also make crucial use of a generalized theory of Tate cohomology recently developed by Mislin.

Applications of a new K-theoretic theorem to soluble group rings

Let R be a ring and let G be a soluble group. In this situation we shall give necessary and sufficient conditions for RG to have a right Artinian right quotient ring. In the course of this work, we shall also consider the Goldie rank problem for soluble groups and record an affirmative answer to the zero divisor conjecture for soluble groups.

Relative ends and duality groups

Abstract We introduce a new algebraic invariant ẽ ( G , S ) for pairs of groups S ⩽ G . It is related to the geometric end invariant of Houghton and Scott, but is more easily accessible to calculation by cohomological methods. We develop various techniques for computing ẽ ( G , S ) when G and S enjoy certain duality properties.

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 complete resolutions

We develop simple criteria for constructing and using complete resolutions for modules over group algebras and strongly group graded rings.

Cohomological finiteness conditions for elementary amenable groups

Abstract It is proved that every elementary amenable group of type FP∞ admits a cocompact classifying space for proper actions.

On a property of fundamental groups of graphs of finite groups

It is shown that for any group G, the Z comprising bounded functions from G to Z is ZF-free for all finite subgroups F of G. This fact is used to establish a property of fundamental groups of graphs of finite groups.

A note on centrality in 3-manifold groups

Centralizers in fundamental groups of 3-manifolds are well understood because of their relationship with Seifert fibre spaces. Jaco and Shalens book [4] provides detailed information about Seifert fibre spaces in 3-manifolds, and consequently about centralizers in their fundamental groups. It is the purpose of this note to record two group-theoretic properties, both easily deduced from results of Jaco and Shalen. Doubtless many other authors could have established the same results had they needed them. Our motivation for writing this paper is that these properties can be used as a basis for group-theoretic proofs of certain fundamental results in 3-manifold theory: Proposition 1 below can be used as a basis for a proof of the Torus Theorem (cf. [8]) and Proposition 2 for the Torus Decomposition Theorem (cf. [9]).

Homological finiteness conditions for modules over strongly group-graded rings

Throughout this paper, k denotes a commutative ring. We will develop a theory of homological finiteness conditions for modules over certain graded k -algebras which generalizes known theory for group algebras. The simplest of our results, Theorem A below, generalizes certain results of Aljadeff and Yi on crossed products of polycyclic-by-finite groups (cf. [1, 11]), but also applies to many other crossed products in cases where little was previously known. Before stating the results, we recall definitions of graded and strongly graded rings. Let G be a monoid. Naively, a G -graded k -algebra is a k -algebra R which admits a k -module decomposition, in such a way that R g R h ⊆ for all g , h ∈ G . If R is a G -graded k -algebra and X is any subset of G , then we write R x for the k -submodule of R supported on X ; that is Note that if H is a submonoid of G then R H is a subalgebra of R .

Powers in Finitely Generated Groups

In this paper we study the set Γn of nth-powers in certain finitely generated groups Γ. We show that, if Γ is soluble or linear, and Γn contains a finite index subgroup, then Γ is nilpotent-by-finite. We also show that, if Γ is linear and Γn has finite index (i.e. Γ may be covered by finitely many translations of Γn), then Γ is soluble-by-finite. The proof applies invariant measures on amenable groups, number-theoretic results concerning the S-unit equation, the theory of algebraic groups and strong approximation results for linear groups in arbitrary characteristic.