Abstract

We outline a proof that if G is a soluble or linear group of type (FP)∞ then G has finite virtual cohomological dimension. The proof depends on hierarchical decompositions of soluble and linear groups and also makes use of a recently discovered generalized Tate cohomology theory. A survey of this complete cohomology is included. The paper concludes with a review of some open problems.

Preface

The first part of this article is based on a lecture delivered at the conference. It concerns the proof that soluble and linear groups of type (FP)∞ have finite vcd. More general results have been published in [21], but in order to make the key new arguments widely accessible I thought it worthwhile going through the special cases again. Several technical problems can be avoided this way, and I hope that this will make for clarity.

At the conference, a number of people asked about the generalized Tate cohomology theory which plays such a crucial and somewhat miraculous role. For this reason I have included a detailed account in §4.

In the last section, some problems and questions are discussed which I did not have time to cover in the lecture. Some of the results in this section have not been published elsewhere.

Introduction

Let G be a group. This paper studies projective resolutions P* ↠ ℤ of the trivial module ℤ over the group ring ℤG.