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 .

Topics & Concepts