Topological 2generation of automorphism groups of countable ultrahomogeneous graphs
Abstract
A countable graph is ultrahomogeneous if every isomorphism between finite induced subgraphs can be extended to an automorphism. Woodrow and Lachlan showed that there are essentially four types of such countably infinite graphs: the random graph; infinite disjoint unions of complete graphs $K_n$ with $n\in \mathbb{N}$ vertices; the $K_n$free graphs; finite unions of the infinite complete graph $K_{\omega}$; and duals of such graphs. The groups $\operatorname{Aut}(\Gamma)$ of automorphisms of such graphs $\Gamma$ have a natural topology, which is compatible with multiplication and inversion, i.e.\ the groups $\operatorname{Aut}(\Gamma)$ are topological groups. We consider the problem of finding minimally generated dense subgroups of the groups $\operatorname{Aut}(\Gamma)$ where $\Gamma$ is ultrahomogeneous. We show that if $\Gamma$ is ultrahomogeneous, then $\operatorname{Aut}(\Gamma)$ has 2generated dense subgroups, and that under certain conditions given $f \in \operatorname{Aut}(\Gamma)$ there exists $g\in \operatorname{Aut}(\Gamma)$ such that the subgroup generated by $f$ and $g$ is dense. We also show that, roughly speaking, $g$ can be chosen with a high degree of freedom. For example, if $\Gamma$ is either an infinite disjoint unions of $K_n$ or a finite union of $K_{\omega}$, then $g$ can be chosen to have any given finite set of orbit representatives.
 Publication:

arXiv eprints
 Pub Date:
 February 2016
 DOI:
 10.48550/arXiv.1602.05766
 arXiv:
 arXiv:1602.05766
 Bibcode:
 2016arXiv160205766J
 Keywords:

 Mathematics  Group Theory
 EPrint:
 Fixed some typos and a couple inaccuracies