Emmanuel Breuillard

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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(More)
We show that any locally compact group G with polynomial growth is weakly commensurable to some simply connected solvable Lie group S, the Lie shadow of G. We then study the shape of large balls and show, generalizing work of P. Pansu, that after a suitable renormalization, they converge to a limiting compact set which can be interpreted geometrically. As a(More)
We show that for every integer d ∈ N, there is N(d) ∈ N such that if K is any field and F is a finite subset of GLd(K), which generates a non amenable subgroup, then F contains two elements, which freely generate a non abelian free subgroup. This improves the original statement of the Tits alternative. It also implies a growth gap and a co-growth gap for(More)
We describe the structure of “K-approximate subgroups” of solvable subgroups of GLn(C), showing that they have a large nilpotent piece. By combining this with the main result of our recent paper on approximate subgroups of torsion-free nilpotent groups [3], we show that such approximate subgroups are efficiently controlled by nilpotent progressions.