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Countable Ultrahomogeneous, Undirected Graphs, R. Woodrow (1980)
COUNTABLE ULTRAHOMOGENEOUS UNDIRECTED GRAPHS1'2
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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.
Forum Mathematicum – de Gruyter
Published: Jul 1, 2017
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