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Forum Mathematicum

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Ed. by Brüdern, Jörg / Cohen, Frederick R. / Droste, Manfred / Duzaar, Frank / Neeb, Karl-Hermann / Noguchi, Junjiro / Shahidi, Freydoon / Sogge, Christopher D. / Wienhard, Anna

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Volume 29, Issue 4 (Jul 2017)

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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

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 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

References

  • [1]

    U. B. Darji and J. D. Mitchell, Highly transitive subgroups of the symmetric group on the natural numbers, Colloq. Math. 112 (2008), no. 1, 163–173. Google Scholar

  • [2]

    U. B. Darji and J. D. Mitchell, Approximation of automorphisms of the rationals and the random graph, J. Group Theory 14 (2011), no. 3, 361–388. CrossrefWeb of ScienceGoogle Scholar

  • [3]

    P. Erdős and A. Rényi, Asymmetric graphs, Acta Math. Acad. Sci. Hungar 14 (1963), 295–315. CrossrefGoogle Scholar

  • [4]

    R. Grząślewicz, Density theorems for measurable transformations, Colloq. Math. 48 (1984), no. 2, 245–250. Google Scholar

  • [5]

    W. Hodges, A Shorter Model Theory, Cambridge University Press, Cambridge, 1997. Google Scholar

  • [6]

    A. S. Kechris, Classical Descriptive Set Theory, Grad. Texts in Math. 156, Springer, Berlin, 1995. Google Scholar

  • [7]

    A. S. Kechris and C. Rosendal, Turbulence, amalgamation, and generic automorphisms of homogeneous structures, Proc. Lond. Math. Soc. (3) 94 (2007), no. 2, 302–350. CrossrefGoogle Scholar

  • [8]

    A. H. Lachlan and R. E. Woodrow, Countable ultrahomogeneous undirected graphs, Trans. Amer. Math. Soc. 262 (1980), no. 1, 51–94. Google Scholar

  • [9]

    H. D. Macpherson, Groups of automorphisms of 0-categorical structures, Quart. J. Math. Oxford Ser. (2) 37 (1986), no. 148, 449–465. Google Scholar

  • [10]

    S. Piccard, Sur les bases du groupe symetrique et du groupe alternant, Math. Ann. 116 (1939), no. 1, 752–767. Google Scholar

  • [11]

    V. S. Prasad, Generating dense subgroups of measure preserving transformations, Proc. Amer. Math. Soc. 83 (1981), no. 2, 286–288. Google Scholar

  • [12]

    S. Solecki, Extending partial isometries, Israel J. Math. 150 (2005), no. 1, 315–331. Google Scholar

  • [13]

    A. Stein, 112-generation of finite simple groups, Beitr. Algebra Geom. 39 (1998), no. 2, 349–358. Google Scholar

  • [14]

    A. J. Woldar, 3/2-generation of the sporadic simple groups, Comm. Algebra 22 (1994), no. 2, 675–685. CrossrefGoogle Scholar

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, ISSN (Online) 1435-5337, ISSN (Print) 0933-7741, DOI: https://doi.org/10.1515/forum-2016-0056.

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