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 for every integer d ∈ N, there is N (d) ∈ N such that if K is any field and F is a finite subset of GL d (K), which generates a non amenable subgroup, then F N (d) 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(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 a global adelic analog of the classical Margulis Lemma from hyperbolic geometry. We introduce a conjugation invariant normalized height h(F) of a finite set of matrices F in SL n (Q) which is the adelic analog of the minimal displacement on a symmetric space. We then show, making use of theorems of Bilu and Zhang on the equidistribu-tion of Galois(More)