Generic elements of a Zariski-dense subgroup form an open subset

Abstract:

Let $G$ be a semisimple algebraic group over a finitely generated field $K$ of characteristic zero, and let $\Gamma \subset G(K)$ be a finitely generated Zariski-dense subgroup. In this paper we prove that the set of $K$-generic elements of $\Gamma$ (whose existence was established earlier by the authors in Existence of irreducible $\mathbb {R}$-regular elements in Zariski-dense subgroups, Math. Res. Lett. $\mathbf {10}$ (2003), no. 1, 21–32, is open in the profinite topology of $\Gamma$. We then extend this result to the fields of positive characteristic, and also prove the existence of generic elements in this case.