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Forum Mathematicum

Managing Editor: Bruinier, Jan Hendrik

Ed. by Cohen, Frederick R. / Droste, Manfred / Darmon, Henri / Duzaar, Frank / Echterhoff, Siegfried / Gordina, Maria / Neeb, Karl-Hermann / Shahidi, Freydoon / Sogge, Christopher D. / Takayama, Shigeharu / Wienhard, Anna

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Volume 29, Issue 4


Topological 2-generation of automorphism groups of countable ultrahomogeneous graphs

Julius Jonušas
  • Mathematical Institute, School of Mathematics and Statistics, University of St Andrews, North Haugh, United Kingdom
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  • Other articles by this author:
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/ James Mitchell
  • Corresponding author
  • Mathematical Institute, School of Mathematics and Statistics, University of St Andrews, North Haugh, United Kingdom
  • Email
  • Other articles by this author:
  • De Gruyter OnlineGoogle Scholar
Published Online: 2016-09-14 | DOI: https://doi.org/10.1515/forum-2016-0056


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 Kn with n vertices, the Kn-free graphs, finite unions of the infinite complete graph Kω, and duals of such graphs. The groups Aut(Γ) of automorphisms of such graphs Γ have a natural topology, which is compatible with multiplication and inversion, i.e. the groups Aut(Γ) are topological groups. We consider the problem of finding minimally generated dense subgroups of the groups Aut(Γ) where Γ is ultrahomogeneous. We show that if Γ is ultrahomogeneous, then Aut(Γ) has 2-generated dense subgroups, and that under certain conditions given fAut(Γ) there exists gAut(Γ) 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 Γ is either an infinite disjoint union of Kn or a finite union of Kω, then g can be chosen to have any given finite set of orbit representatives.

Keywords: Automorphism groups; Fraïssé limits; topological generation; Polish groups

MSC 2010: 54H11; 20B27


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About the article

Received: 2016-02-29

Revised: 2016-06-21

Published Online: 2016-09-14

Published in Print: 2017-07-01

Funding Source: Carnegie Trust for the Universities of Scotland

Award identifier / Grant number: 12820

We thank the Carnegie Trust for the Universities of Scotland for funding the PhD scholarship of J. Jonušas (no. 12820).

Citation Information: Forum Mathematicum, Volume 29, Issue 4, Pages 905–939, ISSN (Online) 1435-5337, ISSN (Print) 0933-7741, DOI: https://doi.org/10.1515/forum-2016-0056.

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