A Strong Tits Alternative
Abstract
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 cogrowth gap for nonamenable 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 nondiscrete subgroups of Lie groups.
 Publication:

arXiv eprints
 Pub Date:
 April 2008
 arXiv:
 arXiv:0804.1395
 Bibcode:
 2008arXiv0804.1395B
 Keywords:

 Mathematics  Group Theory;
 20G25;
 22E40
 EPrint:
 40 pages