A Strong Tits Alternative

By Emmanuel Breuillard. 2008

51 citations recorded by the source corpus.

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.

Subfields: Discrete Mathematics and Combinatorics, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/f06c0f602a65e468f0cd01bd57ba0a808f605f07

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Discrete Mathematics and Combinatorics

In Geometry and Topology

In Mathematical Physics

Citations in the corpus

Approximate Subgroups of Linear Groups (2010, background): “red in [57] in the solvable case and [53] for arbitrary ﬁnitely generated groups. For non-virtually solvable linear groups it is also a immediate consequence of the uniform Tits alternative proved in [12]. 7. A conjecture of Babai and Seress Some applications of Helfgott’s product theorem regarding ”
Recent progress on Bernoulli convolutions (2016, background): “interest. Breuillard [13] has made the following conjecture. Conjecture 3.1 (Breuillard). For every integer d > 0, there is cd > 0 such that ρ(S) ≥ cd or ρ(S) = 0 for any S ⊂ GLd(C). Breuillard [10], [11] and [12] proved this in the important case, when the group generated by S is not virtuall”