Topics in Groups and Geometry : Growth, Amenability, and Random Walks / by Tullio Ceccherini-Silberstein, Michele D'Adderio.

Format:

Book

Author/Creator:

Ceccherini-Silberstein, Tullio, author.

D'Adderio, Michele, author.

Contributor:

Zelmanov, Efim, 1955- writer of foreword.

Series:

Springer Monographs in Mathematics, 2196-9922

Language:

English

Subjects (All):

Group theory.

Associative rings.

Associative algebras.

Geometry.

Probabilities.

Graph theory.

Group Theory and Generalizations.

Associative Rings and Algebras.

Probability Theory.

Graph Theory.

Local Subjects:

Group Theory and Generalizations.

Associative Rings and Algebras.

Geometry.

Probability Theory.

Graph Theory.

Physical Description:

1 online resource (468 pages)

Edition:

1st ed. 2021.

Place of Publication:

Cham : Springer International Publishing : Imprint: Springer, 2021.

Summary:

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.

Contents:

- Foreword

Preface

Part I Algebraic Theory: 1. Free Groups

2. Nilpotent Groups

3. Residual Finiteness and the Zassenhaus Filtration

4. Solvable Groups

5. Polycyclic Groups

6. The Burnside Problem

Part II Geometric Theory: 7. Finitely Generated Groups and Their Growth Functions

8. Hyperbolic Plane Geometry and the Tits Alternative

9. Topological Groups, Lie Groups, and Hilbert Fifth Problem

10. Dimension Theory

11. Ultrafilters, Ultraproducts, Ultrapowers, and Asymptotic Cones

12. Gromov’s Theorem

Part III Analytic and Probabilistic Theory: 13. The Theorems of Polya and Varopoulos

14. Amenability, Isoperimetric Profile, and Følner Functions

15. Solutions or Hints to Selected Exercises

References

Subject Index

Index of Authors.

Other Format:

Print version: Ceccherini-Silberstein, Tullio Topics in Groups and Geometry

ISBN:

3-030-88109-1

OCLC:

1290727081