Topological Methods in Group Theory

In what is set to be a classic in the field, Ross Geoghegan explores the interplay between algebraic topology and the theory of infinite discrete groups. He has written nothing less than a hugely important contribution to the field of topological and geometric group theory.

Aimed at advanced undergraduates and graduate students, this book is about the interplay between algebraic topology and the theory of infinite discrete groups. It is a hugely important contribution to the field of topological and geometric group theory, and is bound to become a standard reference in the field. To keep the length reasonable and the focus clear, the author assumes the reader knows or can easily learn the necessary algebra, but wants to see the topology done in detail. The central subject of the book is the theory of ends. Here the author adopts a new algebraic approach which is geometric in spirit. The author has included material which isn't covered in other textbooks. Much of the material will be of interest to group theorists who would like to know more about the topological side of their subject, as well as manifold topologists looking for basic material on proper homotopy and finite homology.

Covers important topics not covered elsewhere Provides an accessible review of general topology Provides in detail the topological tools needed for group theory Contains more than 40 figures Immense practical value as each chapter includes exercises throughout Self-contained and suitable for graduate courses on algebraic topology or on its applications to group theory

Texte du rabat

Topological Methods in Group Theory is about the interplay between algebraic topology and the theory of infinite discrete groups. The author has kept three kinds of readers in mind: graduate students who have had an introductory course in algebraic topology and who need a bridge from common knowledge to the current research literature in geometric, combinatorial and homological group theory; group theorists who would like to know more about the topological side of their subject but who have been too long away from topology; and manifold topologists, both high- and low-dimensional, since the book contains much basic material on proper homotopy and locally finite homology not easily found elsewhere.

The book focuses on two main themes:

1. Topological Finiteness Properties of groups (generalizing the classical notions of "finitely generated" and "finitely presented");

2. Asymptotic Aspects of Infinite Groups (generalizing the classical notion of "the number of ends of a group").

Illustrative examples treated in some detail include: Bass-Serre theory, Coxeter groups, Thompson groups, Whitehead's contractible 3-manifold, Davis's exotic contractible manifolds in dimensions greater than three, the Bestvina-Brady Theorem, and the Bieri-Neumann-Strebel invariant. The book also includes a highly geometrical treatment of Poincaré duality (via cells and dual cells) to bring out the topological meaning of Poincaré duality groups.

To keep the length reasonable and the focus clear, it is assumed that the reader knows or can easily learn the necessary algebra (which is clearly summarized) but wants to see the topology done in detail. Apart from the introductory material, most of the mathematics presented here has not appeared in book form before.

Résumé

This book is about the interplay between algebraic topology and the theory of infinite discrete groups. It is a hugely important contribution to the field of topological and geometric group theory, and is bound to become a standard reference in the field. To keep the length reasonable and the focus clear, the author assumes the reader knows or can easily learn the necessary algebra, but wants to see the topology done in detail. The central subject of the book is the theory of ends. Here the author adopts a new algebraic approach which is geometric in spirit.

Contenu
Algebraic Topology for Group Theory.- CW Complexes and Homotopy.- Cellular Homology.- Fundamental Group and Tietze Transformation.- Some Techniques in Homotopy Theory.- Elementary Geometric Topology.- Finiteness Properties of Groups.- The Borel Construction and Bass-Serre Theory.- Topological Finiteness Properties and Dimension of Groups.- Homological Finiteness Properties of Groups.- Finiteness Properties of Some Important Groups.- Locally Finite Algebraic Topology for Group Theory.- Locally Finite CW Complexes and Proper Homotopy.- Locally Finite Homology.- Cohomology of CW Complexes.- Topics in the Cohomology of Infinite Groups.- Cohomology of Groups and Ends of Covering Spaces.- Filtered Ends of Pairs of Groups.- Poincaré Duality in Manifolds and Groups.- Homotopical Group Theory.- The Fundamental Group At Infinity.- Higher homotopy theory of groups.- Three Essays.- Three Essays.