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Topological 2-generation of automorphism groups of countable ultrahomogeneous graphs

Topological 2-generation of automorphism groups of countable ultrahomogeneous graphs AbstractA countable graph is ultrahomogeneous if every isomorphism betweenfinite induced subgraphs can be extended to an automorphism. Woodrow andLachlan showed that there are essentially four types of such countablyinfinite graphs: the random graph, infinite disjoint unions of completegraphs Kn{K_{n}}with n∈ℕ{n\in\mathbb{N}}vertices, the Kn{K_{n}}-free graphs, finite unions of theinfinite complete graph Kω{K_{\omega}}, and duals of such graphs. The groupsAut⁡(Γ){\operatorname{Aut}(\Gamma)}of automorphisms of such graphs Γ have a naturaltopology, which is compatible with multiplication and inversion, i.e. thegroups Aut⁡(Γ){\operatorname{Aut}(\Gamma)}are topological groups. We consider the problem offinding minimally generated dense subgroups of the groups Aut⁡(Γ){\operatorname{Aut}(\Gamma)}where Γ is ultrahomogeneous. We show that if Γ isultrahomogeneous, then Aut⁡(Γ){\operatorname{Aut}(\Gamma)}has 2-generated dense subgroups, andthat under certain conditions given f∈Aut⁡(Γ){f\in\operatorname{Aut}(\Gamma)}there exists g∈Aut⁡(Γ){g\in\operatorname{Aut}(\Gamma)}such that the subgroup generated by f and g is dense. Wealso show that, roughly speaking, g can be chosen with a high degree offreedom. For example, if Γ is either an infinite disjoint union ofKn{K_{n}}or a finite union of Kω{K_{\omega}}, then g can be chosen to have anygiven finite set of orbit representatives. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Forum Mathematicum de Gruyter

Topological 2-generation of automorphism groups of countable ultrahomogeneous graphs

Forum Mathematicum , Volume 29 (4): 35 – Jul 1, 2017

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References (14)

Publisher
de Gruyter
Copyright
© 2017 Walter de Gruyter GmbH, Berlin/Boston
ISSN
0933-7741
eISSN
1435-5337
DOI
10.1515/forum-2016-0056
Publisher site
See Article on Publisher Site

Abstract

AbstractA countable graph is ultrahomogeneous if every isomorphism betweenfinite induced subgraphs can be extended to an automorphism. Woodrow andLachlan showed that there are essentially four types of such countablyinfinite graphs: the random graph, infinite disjoint unions of completegraphs Kn{K_{n}}with n∈ℕ{n\in\mathbb{N}}vertices, the Kn{K_{n}}-free graphs, finite unions of theinfinite complete graph Kω{K_{\omega}}, and duals of such graphs. The groupsAut⁡(Γ){\operatorname{Aut}(\Gamma)}of automorphisms of such graphs Γ have a naturaltopology, which is compatible with multiplication and inversion, i.e. thegroups Aut⁡(Γ){\operatorname{Aut}(\Gamma)}are topological groups. We consider the problem offinding minimally generated dense subgroups of the groups Aut⁡(Γ){\operatorname{Aut}(\Gamma)}where Γ is ultrahomogeneous. We show that if Γ isultrahomogeneous, then Aut⁡(Γ){\operatorname{Aut}(\Gamma)}has 2-generated dense subgroups, andthat under certain conditions given f∈Aut⁡(Γ){f\in\operatorname{Aut}(\Gamma)}there exists g∈Aut⁡(Γ){g\in\operatorname{Aut}(\Gamma)}such that the subgroup generated by f and g is dense. Wealso show that, roughly speaking, g can be chosen with a high degree offreedom. For example, if Γ is either an infinite disjoint union ofKn{K_{n}}or a finite union of Kω{K_{\omega}}, then g can be chosen to have anygiven finite set of orbit representatives.

Journal

Forum Mathematicumde Gruyter

Published: Jul 1, 2017

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