A Strong Tits Alternative
Emmanuel Breuillard

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 , there is a constant such that if is any field and is a finite subset of , which generates a non amenable subgroup, then 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 as well as other diophantine properties of non-discrete subgroups of Lie groups.
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Taxonomy
TopicsFinite Group Theory Research · Algebraic Geometry and Number Theory · Advanced Algebra and Geometry
