Topics in Groups and Geometry : Growth, Amenability, and Random Walks

2021 ed.

by Michele D'Adderio , Tullio Ceccherini-Silberstein

Book Details

Format Paperback / Softback

Book Series Springer Monographs in Mathematics

ISBN-10 3030881113

ISBN-13 9783030881115

Edition 2021 ed.

Publisher Springer Nature Switzerland AG

Imprint Springer Nature Switzerland AG

Country of Manufacture GB

Country of Publication GB

Publication Date Nov 24th, 2022

Print length 464 Pages

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It includes classical theorems on nilpotent groups and solvable groups, a fundamental study of the growth of groups, a detailed look at asymptotic cones, and a discussion of related subjects including filters and ultrafilters, dimension theory, hyperbolic geometry, amenability, the Burnside problem, and random walks on groups.

This book provides a detailed exposition of a wide range of topics in geometric group theory, inspired by Gromov''s pivotal work in the 1980s. It includes classical theorems on nilpotent groups and solvable groups, a fundamental study of the growth of groups, a detailed look at asymptotic cones, and a discussion of related subjects including filters and ultrafilters, dimension theory, hyperbolic geometry, amenability, the Burnside problem, and random walks on groups. The results are unified under the common theme of Gromov''s theorem, namely that finitely generated groups of polynomial growth are virtually nilpotent. This beautiful result gave birth to a fascinating new area of research which is still active today.

The purpose of the book is to collect these naturally related results together in one place, most of which are scattered throughout the literature, some of them appearing here in book form for the first time. In this way, the connections between these topics are revealed, providing a pleasant introduction to geometric group theory based on ideas surrounding Gromov''s theorem.

The book will be of interest to mature undergraduate and graduate students in mathematics who are familiar with basic group theory and topology, and who wish to learn more about geometric, analytic, and probabilistic aspects of infinite groups.

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