A Strong Tits Alternative

We show that for every integer d, there is a constant N (d) such that if K is any field and F is a finite subset of GLd (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 gap for non-amenable linear groups, and has consequences about the girth and uniform expansion of small sets in finite subgroups of GLd (Fq) as well as other diophantine properties of non-discrete subgroups of Lie groups.