A Strong Tits Alternative

Details A Strong Tits Alternative A Strong Tits Alternative

2008-04-09

arxiv.org/abs/0804.1395v1

Mathematics

Group Theory

40 pages

Scientific paper

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 $GL_d(K)$, 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 gap for non-amenable linear groups, and has consequences about the girth and uniform expansion of small sets in finite subgroups of $GL_d(\Bbb{F}_q)$ as well as other diophantine properties of non-discrete subgroups of Lie groups.

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