Topological groups and infinite graphs
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).
scientific article; zbMATH DE number 63147
scientific article