Geometry & Topology

Dense embeddings of surface groups

Emmanuel Breuillard, Tsachik Gelander, Juan Souto, and Peter Storm

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We discuss dense embeddings of surface groups and fully residually free groups in topological groups. We show that a compact topological group contains a nonabelian dense free group of finite rank if and only if it contains a dense surface group. Also, we obtain a characterization of those Lie groups which admit a dense faithfully embedded surface group. Similarly, we show that any connected semisimple Lie group contains a dense copy of any fully residually free group.

Article information

Geom. Topol., Volume 10, Number 3 (2006), 1373-1389.

Received: 10 February 2006
Revised: 3 August 2006
Accepted: 18 June 2006
First available in Project Euclid: 20 December 2017

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 22E40: Discrete subgroups of Lie groups [See also 20Hxx, 32Nxx]
Secondary: 20H10: Fuchsian groups and their generalizations [See also 11F06, 22E40, 30F35, 32Nxx]

surface group topological group fully residually free


Breuillard, Emmanuel; Gelander, Tsachik; Souto, Juan; Storm, Peter. Dense embeddings of surface groups. Geom. Topol. 10 (2006), no. 3, 1373--1389. doi:10.2140/gt.2006.10.1373.

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  • M Abert, Y Glasner, Generic groups acting on a regular tree, preprint
  • G Baumslag, On generalised free products, Math. Z. 78 (1962) 423–438
  • E Breuillard, T Gelander, A topological Tits alternative, to appear in Annals of Math.
  • E Breuillard, T Gelander, On dense free subgroups of Lie groups, J. Algebra 261 (2003) 448–467
  • Y de Cornulier, A Mann, Some residually finite groups satisfying laws, to appear in the proceedings of the conference “Asymptotic and Probabilistic Methods in Geometric Group Theory", Geneva (2005)
  • D B A Epstein, Almost all subgroups of a Lie group are free, J. Algebra 19 (1971) 261–262
  • T Gelander, Y Glasner, Countable primitive groups, to appear in Geom. Funct. Anal.
  • T Gelander, A Zuk, Dependence of Kazhdan constants on generating subsets, Israel J. Math. 129 (2002) 93–98
  • V Kaloshin, The existential Hilbert 16-th problem and an estimate for cyclicity of elementary polycycles, Invent. Math. 151 (2003) 451–512
  • A Lubotzky, B Weiss, Groups and expanders, from: “Expanding graphs (Princeton, NJ, 1992)”, DIMACS Ser. Discrete Math. Theoret. Comput. Sci. 10, Amer. Math. Soc., Providence, RI (1993) 95–109
  • G A Margulis, Some remarks on invariant means, Monatsh. Math. 90 (1980) 233–235
  • D Montgomery, L Zippin, Topological transformation groups, Interscience Publishers, New York-London (1955)
  • Z Sela, Diophantine geometry over groups. I. Makanin-Razborov diagrams, Publ. Math. Inst. Hautes Études Sci. (2001) 31–105
  • D Sullivan, For $n>3$ there is only one finitely additive rotationally invariant measure on the $n$-sphere defined on all Lebesgue measurable subsets, Bull. Amer. Math. Soc. $($N.S.$)$ 4 (1981) 121–123
  • J Tits, Free subgroups in linear groups, J. Algebra 20 (1972) 250–270