Algebra Seven: Combinatorial Group Theory. Applications to Geometry

Parshin A. N. (Ed), Shafarevich I. R. (Ed)3-540-51995

This volume of the EMS consists of two parts. The first entitled Combinatorial Group Theory and Fundamental Groups, written by Collins and Zieschang, provides a readable and comprehensive description of that part of group theory which has its roots in topology in the theory of the fundamental group and the theory of discrete groups of transformations. Throughout the emphasis is on the rich interplay between the algebra and the topology and geometry. The second part by Grigorchuk and Kurchanov is a survey of recent work on groups relating to topological manifolds, dealing with equations in groups, particularly in surface groups and free groups, a study in terms of groups of Heegaard decompositions and algorithmic aspects of the Poincaré conjecture, as well as the notion of the growth of groups. The authors have included a list of open problems, some of which have not been considered previously. Both parts contain numerous examples, outlines of proofs and full references to the literature. The book will be very useful as a reference and guide to researchers and graduate students in algebra and topology.