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

This book provides a comprehensive exposition of geometric group theory, inspired by Gromov's pivotal work in the 1980s. It covers classical theorems on nilpotent groups and solvable groups, group growth, asymptotic cones, and related topics such as filters and ultrafilters, dimension theory, hyperbolic geometry, amenability, the Burnside problem, and random walks on groups. The results are unified under Gromov's theorem, which states that finitely generated groups of polynomial growth are virtually nilpotent. The book aims to collect these related results in one place, making it an accessible introduction to geometric group theory for advanced undergraduate and graduate students in mathematics with a background in basic group theory and topology.

Format: Paperback / softback
Length: 464 pages
Publication date: 24 November 2022
Publisher: Springer Nature Switzerland AG

This comprehensive text delves into the realm of geometric group theory, drawing inspiration from Gromov's groundbreaking work in the 1980s. It offers a detailed exposition of a wide array of topics, encompassing classical theorems on nilpotent and solvable groups, a fundamental study of group growth, a thorough exploration of asymptotic cones, and a discussion of related subjects such as filters and ultrafilters, dimension theory, hyperbolic geometry, amenability, the Burnside problem, and random walks on groups. The results are unified under the overarching theme of Gromov's theorem, which asserts that finitely generated groups of polynomial growth possess virtual nilpotency. This remarkable result sparked a captivating new area of research that continues to thrive today. The primary objective of the book is to gather these naturally interconnected results in a single volume, many of which are scattered across various literature sources. By presenting them in this format, the connections between these topics are unveiled, offering a delightful introduction to geometric group theory built upon the foundations of Gromov's theorem.

The book is designed to appeal to advanced undergraduate and graduate students in mathematics with a solid foundation in basic group theory and topology. It provides a comprehensive overview of geometric, analytic, and probabilistic aspects of infinite groups, making it an invaluable resource for those seeking to deepen their understanding of this fascinating field.

Weight: 735g
Dimension: 235 x 155 (mm)
ISBN-13: 9783030881115
Edition number: 1st ed. 2021