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 $G{L}_d(K)$, 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 $G{L}_d({\mathbb{F}}_q)$ as well as other diophantine properties of non-discrete subgroups of Lie groups.

40 pages

References in corpus (1)

• Infinite groups with large balls of torsion elements and small entropy

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• Transience of algebraic varieties in linear groups and application to generic Zariski density
• Positivity of the top Lyapunov exponent for cocycles on semisimple Lie groups over hyperbolic bases
• Strong uniform expansion in $\mathrm{SL}(2,p)$
• Extensions of hyperbolic groups have locally uniform exponential growth
• Non proper HNN extensions and uniform uniform exponential growth