A Strong Tits Alternative

TL;DR

This paper strengthens the Tits alternative by establishing a uniform bound on the power of generating sets in linear groups that guarantees the existence of free subgroups, with implications for growth, expansion, and diophantine properties.

Contribution

It provides a uniform version of the Tits alternative, showing that a fixed power of any generating set yields free subgroups in non-amenable linear groups, improving previous results.

Findings

Existence of a uniform constant N(d) for free subgroup generation

Implications for growth and co-growth gaps in linear groups

Results on girth and expansion in finite linear groups

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 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.