Topics in Groups and Geometry

Growth, Amenability, and Random Walks

Tullio Ceccherini-Silberstein, Michele D'Adderio, Efim Zelmanov

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English

Springer Nature Switzerland AG
24 November 2022

Algebra; Groups & group theory; Geometry; Probability & statistics; Stochastics

Series: Springer Monographs in Mathematics

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.

By: Tullio Ceccherini-Silberstein, Michele D'Adderio
Foreword by: Efim Zelmanov
Imprint: Springer Nature Switzerland AG
Country of Publication: Switzerland
Edition: 2021 ed.
Dimensions: Height: 235mm, Width: 155mm,
Weight: 735g
ISBN: 9783030881115
ISBN 10: 3030881113
Series: Springer Monographs in Mathematics
Pages: 464
Publication Date: 24 November 2022
Audience: Professional and scholarly , Undergraduate
Format: Paperback
Publisher's Status: Active

Tullio Ceccherini-Silberstein graduated from the University of Rome “La Sapienza” in 1990 and obtained his PhD in mathematics at the University of California at Los Angeles in 1994. Since 1997 he has been professor of Mathematical Analysis at the Engineering Department of the Università del Sannio, Benevento (Italy). His main interests include harmonic and functional analysis, geometric and combinatorial group theory, ergodic theory and dynamical systems, and theoretical computer science. He is an editor of the journal Groups, Geometry, and Dynamics, published by the European Mathematical Society, and of the Bulletin of the Iranian Mathematical Society. He has published more than 90 research papers, 9 monographs, and 4 conference proceedings.Michele D'Adderio studied undergraduate mathematics in Bologna and in Rome “La Sapienza”, before obtaining his PhD in mathematics at the University of California at San Diego in 2010. Since 2012 he has been professorat the Mathematics Department of Université Libre de Bruxelles. His main research interests are combinatorial algebra and algebraic combinatorics.

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

“The book under review gives a detailed introduction to several important topics in Geometric Group Theory at a level suitable for advanced undergraduates or graduate students. … The book … is a useful addition to the literature. … Each chapter has a range of exercises at a range of levels … most are independent problems useful for self-study. There are solutions or hints to some of the exercises at the back of the book.” (John M. Mackay, Mathematical Reviews, February, 2024)