Synopses & Reviews
From the reviews of the first printing of this book, published as volume 58 of the Encyclopaedia of Mathematical Sciences: "... This book will be very useful as a reference and guide to researchers and graduate students in algebra and and topology." Acta Scientiarum Mathematicarum, Ungarn, 1994 "... The book under review consists of two monographs on geometric aspects of group theory: Combinatorial group theory and fundamental groups" by D.J.Collins and H.Zieschang ...: "Some problems of group theory related to geometry" by R.I.Grigorchuk and P.F.Kurchanov. ... Together, these two articles form a wide-ranging survey of combinatorial group theory, with emphasis very much on the geometric roots of the subject. This will be a useful reference work for the expert, as well as providing an overview of the subject for the outsider or novice. Many different topics are described and explored, with the main results presented but not proved. This allows the interested reader to get the flavour of these topics without becoming bogged down in detail. Both articles give comprehensive bibliographies, so that it is possible to use this book as the starting point for a more detailed study of a particular topic of interest. ... In summary, a very interesting book! Bulletin of the London Mathematical Society, 1996 "... In both essays the authors give clear and comprehensive definitions, examples and statements (but not proofs) of theorems, so that the book can be understood by a reader with a minimal background in group theory or geometry. Such a reader, needing to find out what is known in this area, will find this a full and accessible store of information." Contemporary Physics, 1994 "...This survey (Part II) presents for the first time that problems in monograph form and by the way offers a unifying treatment of the various approaches to their solutions, as far as they are known, together with hints to open problems. A titbit for every interested reader!" Monatshefte für Mathematik, 1995
Synopsis
This book consists of two parts. The first part provides a comprehensive description of that part of group theory which has its roots in topology. The second more advanced part deals with recent work on groups relating to topological manifolds. It is a valuable guide to research in this field. The text contains numerous examples, sketches of proofs and open problems.
Synopsis
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 PoincarA(c) 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.
Synopsis
From the reviews: "... The book under review consists of two monographs on geometric aspects of group theory ... Together, these two articles form a wide-ranging survey of combinatorial group theory, with emphasis very much on the geometric roots of the subject. This will be a useful reference work for the expert, as well as providing an overview of the subject for the outsider or novice. Many different topics are described and explored, with the main results presented but not proved. This allows the interested reader to get the flavour of these topics without becoming bogged down in detail. Both articles give comprehensive bibliographies, so that it is possible to use this book as the starting point for a more detailed study of a particular topic of interest. ..." Bulletin of the London Mathematical Society, 1996
Table of Contents
Part I: Combinatorial Group Theory and Fundamental Groups by D.J. Collins and H. Zieschang.- Part II: Some Questions of Group Theory Related to Geometry by R.I. Grigorchuk and P.F. Kurchanov.