Abstract

This article is an edited account of the four lectures that I gave at the Groups St Andrews meeting in Bath during the summer of 1997. The aim of these lectures was to introduce an audience of group theorists, with varying backgrounds, to the role that non-positive curvature plays in the theory of discrete groups. A few new results are included, but basically this is an expository article aimed as non-experts. These notes do not constitute a comprehensive survey of curvature in group theory. Nevertheless, I hope that they give the reader a substantial and enticing taste of this active area of research.

Introduction

In the last fifteen years the close connection between geometry and combinatorial group theory, which was at the heart of the pioneering work of Dehn [52], has re-emerged as a central theme in the study of infinite groups. Thus combinatorial group theory has been joined by (and to a large extent has merged with) what has become known as geometric group theory — this is the broad context of these lectures. My basic goal will be to illustrate how various notions of non-positive curvature can be used to illuminate and solve a range of group theoretic problems. I shall also attempt to illustrate how group theory can serve as a potent tool for exploring the geometry of non-positively curved spaces.