This book is a collection of articles addressing a range of topics in the theory of discrete groups; for the most part the papers represent talks delivered at a conference held at the University of Birmingham in January 1991 to mark the retirement of A. M. Macbeath from his chair at the University of Pittsburgh.

The central theme of the volume is the study of groups from a geometric point of view. Of course, the geometric aspect takes many forms; thus one may find a group which operates on Euclidean or hyperbolic space rubbing shoulders with free groups or some generalisation studied with the help of graphs or homological algebra. Groups with presentation also relate to abstract or formal geometry: in recent years the study of groups which act on trees has brought algebraic structures back into the geometric fold, and the seminal idea that a countable group itself carries a geometric essence has reinforced this return to geometry within rather than through group theory, giving Klein's Programme a fresh cutting edge.

A major part of group theory today relates directly to explicit algebraic or geometric objects — permutation groups, Coxeter groups and discrete subgroups of Lie groups are prominent — and one of the strengths of this field lies in the wealth of fascinating interactions with complex analysis and low dimensional topology. The serious study of discrete groups via combinatorial techniques began in hyperbolic space with Poincare and Dehn, and the reader will find here many echoes of their original ideas and interests.