Geometric Group Theory: An Introduction

Inspired by classical geometry, geometric group theory has in turn provided a variety of applications to geometry, topology, group theory, number theory and graph theory. This carefully written textbook provides a rigorous introduction to this rapidly evolving field whose methods have proven to be powerful tools in neighbouring fields such as geometric topology.Geometric group theory is the study of finitely generated groups via the geometry of their associated Cayley graphs. It turns out that the essence of the geometry of such groups is captured in the key notion of quasi-isometry, a large-scale version of isometry whose invariants include growth types, curvature conditions, boundary constructions, and amenability.This book covers the foundations of quasi-geometry of groups at an advanced undergraduate level. The subject is illustrated by many elementary examples, outlooks on applications, as well as an extensive collection of exercises.

Sommario

1 Introduction.- Part I Groups.- 2 Generating groups.- Part II Groups > Geometry.- 3 Cayley graphs.- 4 Group actions.- 5 Quasi-isometry.- Part III Geometry of groups.- 6 Growth types of groups.- 7 Hyperbolic groups.- 8 Ends and boundaries.- 9 Amenable groups.- Part IV Reference material.- A Appendix.- Bibliography.- Indices.

Relazione

"The structure of the chapters can make the reader independent, thus the book can be used 'outside of the classroom' for self-teaching by both young researchers and experienced scholars. The book is well written ... . it is ready to fill a gap in the literature for such an interesting and active branch of mathematics." (Dimitrios Varsos, zbMATH 1426.20001, 2020)

Dettagli sul prodotto

• Autori: Clara Löh
• Editore: Springer International Publishing AG
• Lingue: Inglese
• Formato: Tascabile
• Pubblicazione: 19.01.2018
• EAN: 9783319722535
• ISBN: 978-3-31-972253-5
• Dimensioni: 155 mm x 235 mm x 23 mm
• Serie: Universitext; Universitext
• Categorie: Scienze naturali, medicina, informatica, tecnica > Matematica > Aritmetica, algebra; B, Kombinatorik und Graphentheorie, Differentielle und Riemannsche Geometrie, Topologie, Nichteuklidische Geometrie, Group Theory, Mathematics and Statistics, Differential Geometry, Combinatorics & graph theory, Manifolds and Cell Complexes (incl. Diff.Topology), Manifolds (Mathematics), Analytic geometry, Manifolds and Cell Complexes, Complex manifolds, Analytic topology, Group Theory and Generalizations, Differential & Riemannian geometry, Graph Theory, Hyperbolic Geometry, Non-Euclidean geometry