Book contents
- Frontmatter
- Contents
- Foreword
- Participants
- On bounded languages and the geometry of nilpotent groups
- Finitely presented groups and the finite generation of exterior powers
- Semigroup presentations and minimal ideals
- Generalised trees and Λ-trees
- The mathematician who had little wisdom: a story and some mathematics
- Palindromic automorphisms of free groups
- A Freiheitssatz for certain one-relator amalgamated products
- Isoperimetric functions of groups and exotic cohomology
- Some embedding theorems and undecidability questions for groups
- Some results on bounded cohomology
- On perfect subgroups of one-relator groups
- Weight tests and hyperbolic groups
- A non-residually finite, relatively finitely presented group in the variety N2A
- Hierarchical decompositions, generalized Tate cohomology, and groups of type (FP)∞
- Tree-lattices and lattices in Lie groups
- Generalisations of Fibonacci numbers, groups and manifolds
- Knotted surfaces in the 4-sphere with no minimal Seifert manifolds
- The higher geometric invariants of modules over Noetherian group rings
- On calculation of width in free groups
- Hilbert modular groups and isoperimetric inequalities
- On systems of equations in free groups
- Cogrowth and essentiality in groups and algebras
- Regular geodesic languages for 2-step nilpotent groups
- Finding indivisible Nielsen paths for a train track map
- More on Burnside's problem
- Problem Session
Hierarchical decompositions, generalized Tate cohomology, and groups of type (FP)∞
Published online by Cambridge University Press: 05 April 2013
Book contents
- Frontmatter
- Contents
- Foreword
- Participants
- On bounded languages and the geometry of nilpotent groups
- Finitely presented groups and the finite generation of exterior powers
- Semigroup presentations and minimal ideals
- Generalised trees and Λ-trees
- The mathematician who had little wisdom: a story and some mathematics
- Palindromic automorphisms of free groups
- A Freiheitssatz for certain one-relator amalgamated products
- Isoperimetric functions of groups and exotic cohomology
- Some embedding theorems and undecidability questions for groups
- Some results on bounded cohomology
- On perfect subgroups of one-relator groups
- Weight tests and hyperbolic groups
- A non-residually finite, relatively finitely presented group in the variety N2A
- Hierarchical decompositions, generalized Tate cohomology, and groups of type (FP)∞
- Tree-lattices and lattices in Lie groups
- Generalisations of Fibonacci numbers, groups and manifolds
- Knotted surfaces in the 4-sphere with no minimal Seifert manifolds
- The higher geometric invariants of modules over Noetherian group rings
- On calculation of width in free groups
- Hilbert modular groups and isoperimetric inequalities
- On systems of equations in free groups
- Cogrowth and essentiality in groups and algebras
- Regular geodesic languages for 2-step nilpotent groups
- Finding indivisible Nielsen paths for a train track map
- More on Burnside's problem
- Problem Session
Summary
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.
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- Combinatorial and Geometric Group Theory, Edinburgh 1993 , pp. 190 - 216Publisher: Cambridge University PressPrint publication year: 1994
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