Abstract: I will describe a joint work with E. Breuillard.

In his celebrated 1972 paper, J. Tits proved a basic and important dichotomy for linear groups, known today as the Tits Alternative: A finitely generated linear group is either virtually solvable or contains a non-abelian free subgroup.

We proved two generalizations of Tits' theorem, 1. topological and 2. effective:

1. A topological Tits alternative: Let k be a local field and G is a subgroup GL(n,K). Then either G contains a relatively open solvable subgroup or it contains a relatively dense free subgroup.

For k=R it answers a question of Carriere and Ghys and provides a short proof for a conjecture of Connes and Sullivan on amenable actions which was first proved by Zimmer by other methods.

For k non-Archimedean it implies a conjecture of Dixon, Pyber, Seress and Shalev concerning the profinite completion of linear groups.

As a further application, we settle a conjecture of Carriere, that the growth of the leaves in any Riemannian foliation on a compact manifold is either polynomial or exponential.

2. An effective version: Let G be a non-virtually solvable finitely generated linear group (over a field of char 0), then there is a constant M such that for any generating set S of G, the M'th ball S^M contains two elements which generate a non-abelian free group.

This result improves the recent theorem of Eskin-Mozes-Oh that such groups have uniform exponential growth (our proof uses their methods as well as some new methods).

As an application we obtain that non-amenable linear groups are uniformly non-amenable.