On dense free subgroups of Lie groups

By Emmanuel Breuillard, Tsachik Gelander. 2002

151 citations recorded by the source corpus.

Abstract We give a method for constructing dense and free subgroups in real Lie groups. In particular we show that any dense subgroup of a connected semisimple real Lie group G contains a free group on two generators which is still dense in G , and that any finitely generated dense subgroup in a connected non-solvable Lie group H contains a dense free subgroup of rank ⩽2·dim H . This answers a question of Carriere and Ghys, and it gives an elementary proof to a conjecture of Connes and Sullivan on amenable actions, which was first proved by Zimmer.

Subfields: Applied Mathematics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/6ab03a3eddf97406464909823e0a505a817a8b69

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In Applied Mathematics

In Geometry and Topology

Citations in the corpus

Random subgroups of linear groups are free (2010, methodology): “The following lemma taken from [BG03] shows that a large ratio between a1(g) and a2(g) implies contraction.”
Hyperbolic structures on groups (2017, background): “y dense subgroup of PSL(2;R) is coarsely transitive on pairs of equidistant points. This yields examples of minimal hyperbolic structures on dense subgroups of PSL(2;R), e.g., on PSL(2;Q) or F 2 (see [22,18] for examples of dense free subgroups of PSL(2;R)). It is clear that every lineal action sati”
A topological Tits alternative (2004, methodology): “However, in [4] and [35], a lot of freedom was possible in the choice of K and in the choice of the representation into GLn(K).”
Strongly dense free subgroups of semisimple algebraic groups (2010, background): “Gelander and the first author showed in [6, 7] that every subgroup of a semisimple real Lie group which is dense in the Euclidean topology contains a free subgroup on 2 generators that is again dense in the Euclidean topology.”
Uniform independence in linear groups (2006, background): “For a more exhaustive and detailed study of the dynamical properties of projective transformations we refer the reader to [[6], Section 3] and [[7], Section 3].”