Topics in Groups and Geometry

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.

Provides an introduction to geometric group theory based on the unifying theme of Gromov's theorem Shows the connections between a wide range of topics in geometric group theory Collects together, for the first time, results previously scattered throughout the literature With a Foreword by Efim I. Zelmanov

Auteur

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 professor at the Mathematics Department of Université Libre de Bruxelles. His main research interests are combinatorial algebra and algebraic combinatorics.

Résumé

"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)

Contenu

• 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. Ultralters, 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 Prole, and Følner Functions.- 15. Solutions or Hints to Selected Exercises.- References.- Subject Index.- Index of Authors.