Topological Methods in Group Theory (Graduate Texts in Mathematics, Band 243) - Softcover

Über die Autorin bzw. den Autor

Matthias Beck received his initial training in mathematics in Würzburg, Germany, received his Ph.D. in mathematics from Temple University, and is now associate professor of mathematics at San Francisco State University. He is the recipient of the 2013 MAA Haimo Award for Distinguished College or University Teaching of Mathematics. He is the author of a previously published Springer book, Computing the Continuous Discretely (with Sinai Robins). Ross Geoghegan received his initial training in mathematics in Dublin, Ireland, received his Ph.D. in mathematics from Cornell University, and is now professor of mathematics at the State University of New York at Binghamton. He is the author of a previously published Springer book, Topological Methods in Group Theory.

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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.