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Geometric Group Theory

An Introduction

  • Textbook
  • © 2017

Overview

Authors:
  1. Clara Löh

    1. Fakultät für Mathematik, Universität Regensburg , Regensburg, Germany

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  • Features more than 250 exercises of varying difficulty including programming tasks
  • Introduces the key notions from quasi-geometry, such as growth, hyperbolicity, boundary constructions and amenability
  • Assumes only a basic background in group theory, metric spaces and point-set topology

Part of the book series: Universitext (UTX)

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Table of contents (9 chapters)

  1. Front Matter

    Pages i-xi
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  2. Introduction

    • Clara Löh
    Pages 1-5

  3. Groups

    1. Front Matter

      Pages 7-7
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    2. Generating groups

      • Clara Löh
      Pages 9-49

  4. Groups → Geometry

    1. Front Matter

      Pages 51-51
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    2. Cayley graphs

      • Clara Löh
      Pages 53-74

    3. Group actions

      • Clara Löh
      Pages 75-114

    4. Quasi-isometry

      • Clara Löh
      Pages 115-163

  5. Geometry of groups

    1. Front Matter

      Pages 165-165
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    2. Growth types of groups

      • Clara Löh
      Pages 167-202

    3. Hyperbolic groups

      • Clara Löh
      Pages 203-256

    4. Ends and boundaries

      • Clara Löh
      Pages 257-287

    5. Amenable groups

      • Clara Löh
      Pages 289-315

  6. Back Matter

    Pages 317-389
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Keywords

About this book

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.

Reviews

“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)

Authors and Affiliations

  • Fakultät für Mathematik, Universität Regensburg , Regensburg, Germany

    Clara Löh

About the author

Clara Löh is Professor of Mathematics at the University of Regensburg, Germany. Her research focuses on the interaction between geometric topology, geometric group theory, and measurable group theory. This includes cohomological, geometric, and combinatorial methods. 

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