A Strong Tits Alternative

2008/04/09 by Breuillard, Emmanuel
#20G25 #22E40 #FOS: Mathematics #Group Theory (math.GR)

paper · doi:10.48550/arxiv.0804.1395

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(\BbbF_q) as well as other diophantine properties of non-discrete subgroups of Lie groups.

Related

• A topological Tits alternative
• The Tits Alternative for two-dimensional Artin groups and Wise's Power Alternative
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• Solvable subgroups of Out(F_n) are virtually abelian
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