Geometric group theory. An introduction
type: Item
instance of: scholarly article
title: Geometric group theory. An introduction
zbMATH Open document ID: 1426.20001
DOI: 10.1007/978-3-319-72254-2
author: Clara Löh
published in: Universitext
publication date: 2018-01-09T00:00:00Z
review text: The book consists from 9 chapters in three parts and an appendix (in Part IV). Part I (Chapter 2) begins with the definition of groups, in few pages running the basic notions to end up to its aim, the definition of a free group and the resulting notion of the presentation of a group with generators and relations. The chapter concludes with a first development of amalgamated free products and HNN-extensions. A fundamental question of geometric group theory is how groups can be viewed as geometric objects. Part II (Chapters 3, 4 and 5) is devoted to study this question. In Chapter 3, the notion of the Cayley graph of a (finitely generated) group is developed and, apart the definitions and the properties of Cayley graphs, the chapter focuses to the tree of a free group. In Chapter 4, the action of a group on graphs is developed and, more specially, the (natural) action of a group on its Cayley graph is studied. This leads to the characterization of a free group by its (free) action on a tree. Applications of this characterization are the Nielsen-Schreier theorem and the corollaries of this theorem. Another (sufficient) criterion (the ping-pong lemma) for the freeness of a group is proved and as an application it is proved that linear groups contain free groups as subgroups. Finally, a sketch of a proof of the Tits alternative is given. If \(S\) is a generating set of a group \(G\), then the associated Cayley graph \(\mathrm{Cay}(G,\,S)\) is endowed with a metric (the word metric with respect to the generating set \(S\)). This metric depends on the set \(S\). So, since the group would be considered as geometric object independent of the chosen generating set, this difficulty must be overcome. This is obtained by passing to large scale geometry. Chapter 5 is devoted to this purpose. The notions of isometry, bilipschitz equivalence and quasi-isometry of metric spaces are developed and are applied to characterize groups up to bilipschitz equivalence and quasi-isometry. Her…
reviewed by: Dimitrios Varsos
Mathematics Subject Classification ID: 20-01
Mathematics Subject Classification ID: 20F65
Mathematics Subject Classification ID: 20F67
Mathematics Subject Classification ID: 05C25
zbMATH DE Number: 6823413
zbMATH Keywords: geometric group theory
zbMATH Keywords: cayley graphs
zbMATH Keywords: quasi-isometry
zbMATH Keywords: growth of groups
zbMATH Keywords: hyperbolic groups
zbMATH Keywords: ends of groups
zbMATH Keywords: amenable groups
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1007/978-3-319-72254-2
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description: scientific article; zbMATH DE number 6823413