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

Abstract

Let G be a semi-simple algebraic group over a finitely generated field K of characteristic zero, and let Γ< G(K) be a finitely generated Zariski-dense subgroup. In this note we prove that the set of K-generic elements of Γ(whose existence was established earlier in [9]) is open in the profinite topology of Γ. We then extend this result to the fields of positive characteristic, and also prove the existence of generic elements in this case.

We now prove the existence of generic elements in arbitrary Zariski-dense subsemigroups