Startseite Topological 2-generation of automorphism groups of countable ultrahomogeneous graphs
Artikel
Lizenziert
Nicht lizenziert Erfordert eine Authentifizierung

Topological 2-generation of automorphism groups of countable ultrahomogeneous graphs

  • Julius Jonušas und James Mitchell EMAIL logo
Veröffentlicht/Copyright: 14. September 2016
Forum Mathematicum
Aus der Zeitschrift Forum Mathematicum Band 29 Heft 4

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∈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 f∈Aut⁡(Γ) there exists g∈Aut⁡(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

Communicated by Manfred Droste

Funding source: Carnegie Trust for the Universities of Scotland

Award Identifier / Grant number: 12820

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

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. 10.4064/cm112-1-9Suche in 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. 10.1515/jgt.2010.057Suche in Google Scholar

[3] P. Erdős and A. Rényi, Asymmetric graphs, Acta Math. Acad. Sci. Hungar 14 (1963), 295–315. 10.1007/BF01895716Suche in Google Scholar

[4] R. Grząślewicz, Density theorems for measurable transformations, Colloq. Math. 48 (1984), no. 2, 245–250. 10.4064/cm-48-2-245-250Suche in Google Scholar

[5] W. Hodges, A Shorter Model Theory, Cambridge University Press, Cambridge, 1997. Suche in Google Scholar

[6] A. S. Kechris, Classical Descriptive Set Theory, Grad. Texts in Math. 156, Springer, Berlin, 1995. 10.1007/978-1-4612-4190-4Suche in 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. 10.1112/plms/pdl007Suche in Google Scholar

[8] A. H. Lachlan and R. E. Woodrow, Countable ultrahomogeneous undirected graphs, Trans. Amer. Math. Soc. 262 (1980), no. 1, 51–94. 10.1090/S0002-9947-1980-0583847-2Suche in Google Scholar

[9] H. D. Macpherson, Groups of automorphisms of ℵ0-categorical structures, Quart. J. Math. Oxford Ser. (2) 37 (1986), no. 148, 449–465. 10.1093/qmath/37.4.449Suche in Google Scholar

[10] S. Piccard, Sur les bases du groupe symetrique et du groupe alternant, Math. Ann. 116 (1939), no. 1, 752–767. 10.1007/BF01597389Suche in Google Scholar

[11] V. S. Prasad, Generating dense subgroups of measure preserving transformations, Proc. Amer. Math. Soc. 83 (1981), no. 2, 286–288. 10.1090/S0002-9939-1981-0624915-2Suche in Google Scholar

[12] S. Solecki, Extending partial isometries, Israel J. Math. 150 (2005), no. 1, 315–331. 10.1007/BF02762385Suche in Google Scholar

[13] A. Stein, 1⁤12-generation of finite simple groups, Beitr. Algebra Geom. 39 (1998), no. 2, 349–358. Suche in Google Scholar

[14] A. J. Woldar, 3/2-generation of the sporadic simple groups, Comm. Algebra 22 (1994), no. 2, 675–685. 10.1080/00927879408824867Suche in Google Scholar

Received: 2016-2-29
Revised: 2016-6-21
Published Online: 2016-9-14
Published in Print: 2017-7-1

© 2017 Walter de Gruyter GmbH, Berlin/Boston

Artikel in diesem Heft

  1. Frontmatter
  2. Hurewicz fibrations, almost submetries and critical points of smooth maps
  3. Lower bounds for regular genus and gem-complexity of PL 4-manifolds
  4. Diffusion semigroup on manifolds with time-dependent metrics
  5. Blow-up algebras, determinantal ideals, and Dedekind–Mertens-like formulas
  6. A computational approach to Milnor fiber cohomology
  7. Metrical universality for groups
  8. The second and third moment of L(1/2,χ) in the hyperelliptic ensemble
  9. Slender domains and compact domains
  10. Topological 2-generation of automorphism groups of countable ultrahomogeneous graphs
  11. A note on local Hardy spaces
  12. On the value group of a model of Peano Arithmetic
  13. Extremal values of the (fractional) Weinstein functional on the hyperbolic space
  14. Maximal subsemigroups of some semigroups of order-preserving mappings on a countably infinite set
  15. Unitals in shift planes of odd order
https://doi.org/10.1515/forum-2016-0056
Schlagwörter für diesen Artikel
Automorphism groups; Fraïssé limits; topological generation; Polish groups

Artikel in diesem Heft

  1. Frontmatter
  2. Hurewicz fibrations, almost submetries and critical points of smooth maps
  3. Lower bounds for regular genus and gem-complexity of PL 4-manifolds
  4. Diffusion semigroup on manifolds with time-dependent metrics
  5. Blow-up algebras, determinantal ideals, and Dedekind–Mertens-like formulas
  6. A computational approach to Milnor fiber cohomology
  7. Metrical universality for groups
  8. The second and third moment of L(1/2,χ) in the hyperelliptic ensemble
  9. Slender domains and compact domains
  10. Topological 2-generation of automorphism groups of countable ultrahomogeneous graphs
  11. A note on local Hardy spaces
  12. On the value group of a model of Peano Arithmetic
  13. Extremal values of the (fractional) Weinstein functional on the hyperbolic space
  14. Maximal subsemigroups of some semigroups of order-preserving mappings on a countably infinite set
  15. Unitals in shift planes of odd order
Jetzt anmelden und 20% sparen
Institutioneller Zugang
Wie funktioniert Authentifizierung?
Haben Sie eine Idee, wie wir unsere Website verbessern können?
Bitte schreiben Sie uns.
© 2025 De Gruyter Brill
Heruntergeladen am 1.10.2025 von https://www.degruyterbrill.com/document/doi/10.1515/forum-2016-0056/html
Button zum nach oben scrollen