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# Forum Mathematicum

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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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/ James Mitchell
• Corresponding author
• Mathematical Institute, School of Mathematics and Statistics, University of St Andrews, North Haugh, United Kingdom
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• Other articles by this author:
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 ${K}_{n}$ with $n\in ℕ$ vertices, the ${K}_{n}$-free graphs, finite unions of the infinite complete graph ${K}_{\omega }$, and duals of such graphs. The groups $\mathrm{Aut}\left(\mathrm{\Gamma }\right)$ of automorphisms of such graphs Γ have a natural topology, which is compatible with multiplication and inversion, i.e. the groups $\mathrm{Aut}\left(\mathrm{\Gamma }\right)$ are topological groups. We consider the problem of finding minimally generated dense subgroups of the groups $\mathrm{Aut}\left(\mathrm{\Gamma }\right)$ where Γ is ultrahomogeneous. We show that if Γ is ultrahomogeneous, then $\mathrm{Aut}\left(\mathrm{\Gamma }\right)$ has 2-generated dense subgroups, and that under certain conditions given $f\in \mathrm{Aut}\left(\mathrm{\Gamma }\right)$ there exists $g\in \mathrm{Aut}\left(\mathrm{\Gamma }\right)$ 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 ${K}_{n}$ or a finite union of ${K}_{\omega }$, then g can be chosen to have any given finite set of orbit representatives.

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.

• [3]

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

• [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.

• [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 ${\mathrm{\aleph }}_{0}$-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, $1⁤\frac{1}{2}$-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.

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,

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