Topological groups and infinite graphs
type: Item
instance of: scholarly article
title: Topological groups and infinite graphs
zbMATH Open document ID: 0757.05060
DOI: 10.1016/0012-365X(91)90348-6
author: Wolfgang Woess
published in: Discrete Mathematics
publication date: 1992-09-27T00:00:00Z
review text: The author shows how structural properties of infinite, locally finite, vertex-symmetric graphs can be related to the structure of the automorphism groups of the graphs. After a brief introduction to the structure of topological groups, Section 3 is dedicated to a short proof of a result of \textit{V. I. Trofimov} [Math. USSR, Sb. 51, 405-417 (1985; Zbl 0565.05035), Theorem 2]: If the graph \(\Gamma(V,E)\) has polynomial growth and \(G\) is a group of automorphisms of \(\Gamma\) which acts transitively on \(V\), then there is an imprimitivity system \(\sigma\) of \(G\) on \(V\) with finite blocks, such that \(G^ \sigma\) is a finitely generated nilpotent-by-finite group with finite vertex stabilizers on \(\Gamma^ \sigma\). In Section 4, using results of Trofimov, the following extension of a result of \textit{Ch. D. Godsil}, \textit{W. Imrich}, \textit{N. Seifert}, \textit{M. W. Watkins} and the author [Graphs Combin. 5, 333-338 (1989; Zbl 0714.05029)] is proved: Let \(\Gamma\) be a vertex- symmetric graph; then \(B_ 0(\Gamma)\) is a closed normal subgroup of \(\text{Aut}(\Gamma)\) and \(B(\Gamma)/B_ 0(\Gamma)\) is a torsion free discrete abelian group (where \(B(\Gamma)\) represents the group of bounded automorphisms of \(\Gamma\) while \(B_ 0(\Gamma)\) is the subgroup of bounded automorphisms with finite order). The paper ends with three problems on infinite graphs, two of which have recently been solved (according to notes added in proof).
reviewed by: Richard G. Wilson
Mathematics Subject Classification ID: 05C25
zbMATH DE Number: 63147
zbMATH Keywords: automorphism group
zbMATH Keywords: locally finite graph
zbMATH Keywords: polynomial growth
zbMATH Keywords: \(FC^ -\)- groups
zbMATH Keywords: expanders
zbMATH Keywords: amenability
zbMATH Keywords: vertex-symmetric graph
MaRDI profile type: Publication
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full work available at URL: 0012-365x(91)90348-6
OpenAlex ID: W2050212605
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description: scientific article; zbMATH DE number 63147