Hostname: page-component-745bb68f8f-b6zl4 Total loading time: 0 Render date: 2025-02-07T01:00:17.135Z Has data issue: false hasContentIssue false
  • English
  • Français

Rooted trees, strong cofinality and ample generics

Published online by Cambridge University Press:  01 October 2012

MACIEJ MALICKI
MACIEJ MALICKI*
Affiliation:
Institute of Mathematics, Polish Academy of Sciences, Sniadeckich 8, 00-956, Warsaw, Poland, and Lazarski University, Swieradowska 43, 02-662, Warsaw, Poland. e-mail: mamalicki@gmail.com
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

We characterize those countable rooted trees with non-trivial components whose full automorphism group has uncountable strong cofinality, and those whose full automorphism group contains an open subgroup with ample generics.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 154 , Issue 2 , March 2013 , pp. 213 - 223
Copyright
Copyright © Cambridge Philosophical Society 2012

References

REFERENCES

[1]Bass, H. and Lubotzky, A.Rigidity of group actions on locally finite trees. Proc. London Math. Soc. (3) 69 (1994), no. 3, 541–575.CrossRefGoogle Scholar
[2]Bergman, G. M.Generating infinite symmetric groups. Bull. London Math. Soc. 38 (2006), 429–440.CrossRefGoogle Scholar
[3]Dixon, J. D., Neumann, P. M. and Thomas, S.Subgroups of small index in infinite symmetric groups. Bull. London Math. Soc. 18 (1986), no. 6, 580–586.CrossRefGoogle Scholar
[4]Droste, M. and Göbel, R.Uncountable cofinalities of permutation groups. J. London Math. Soc. (2) 71 (2005), 335–344.CrossRefGoogle Scholar
[5]Droste, M. and Holland, W. C.Generating automorphism groups of chains. Forum Math. 17 (2005), 699–710.CrossRefGoogle Scholar
[6]Forester, M.Deformation and rigidity of simplicial group actions on trees. Geom. Topol. 6 (2002), 219–267.CrossRefGoogle Scholar
[7]Gao, S.Invariant Descriptive Set Theory (CRC Press, 2009).Google Scholar
[8]Ch, W.Holland. The characterization of generalized wreath products. J. Algebra 13 (1969), 152–172.Google Scholar
[9]Hodges, W., Hodkinson, I., Lascar, D. and Shelah, S.The small index property for ω-stable ω-categorical structures and for the random graph. J. London Math. Soc. (2) 48 (1993), 204–218.CrossRefGoogle Scholar
[10]Kechris, A. and Rosendal, C.Turbulence, amalgamation, and generic automorphisms of homogeneous structures. Proc. Lond. Math. Soc. (3) 94 (2007), no. 2, 302–350.CrossRefGoogle Scholar
[11]Psaltis, P.A rigidity theorem for automorphism groups of trees. Israel J. Math. 163 (2008), 345–367.CrossRefGoogle Scholar
[12]Serre, J.-PTrees (Springer, New York, 2003).Google Scholar