Growth in Chevalley groups relatively to parabolic subgroups and some applications

Abstract

Given a Chevalley group $\mathbf{G}(q)$ and a parabolic subgroup $P\subset \mathbf{G}(q)$, we prove that for any set $A$ there is a certain growth of $A$ relatively to $P$, namely, either $AP$ or $PA$ is much larger than $A$. Also, we study a question about the intersection of $A^n$ with parabolic subgroups $P$ for large $n$. We apply our method to obtain some results on a modular form of Zaremba’s conjecture from the theory of continued fractions, and make the first step towards Hensley's conjecture about some Cantor sets with Hausdorff dimension greater than $1/2$.