Let Γ be a subgroup of GL d (Z[1/q0]) generated by a finite symmetric set S. For an integer q, denote by πq the projection mapWe prove that the Cayley graphs of πq(Γ) with respect to the generating sets πq(S) form a family of expanders when q ranges over square-free integers with large prime divisors if and only if the connected component of the Zariski-closure of Γ is perfect, i.e. it has no nontrivial Abelian quotients.

Abstract. We introduce a novel notion of local spectral gap for general, possibly infinite, measure preserving actions. We establish local spectral gap for the left translation action Γ G, whenever Γ is a dense subgroup generated by algebraic elements of an arbitrary connected simple Lie group G. This extends to the non-compact setting works of Bourgain and Gamburd [BG06, BG10], and Benoist and de Saxcé [BdS14]. We present several applications to the Banach-Ruziewicz problem, orbit equivalence rigidity, continuous and monotone expanders, and bounded random walks on G. In particular, we prove that, up to a multiplicative constant, the Haar measure is the unique Γ-invariant finitely additive measure defined on all bounded measurable subsets of G.

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Abstract. Let G be a semisimple Lie group without compact factors, Γ be an irreducible lattice in G. In the first part of the article we give the necessary and sufficient condition under which a sequence of translates of probability "horospherical measures" is convergent. And the limiting measure is also determined when it is convergent (see Theorems 1 and 2 for the precise statements). In the second part, two applications are presented. The first one is of geometric nature and the second one gives an alternative way to count the number of rational points on a flag variety.

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