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

TL;DR

The paper proves that in finitely generated Zariski-dense subgroups of semi-simple algebraic groups over characteristic zero fields, the set of K-generic elements is open in the profinite topology, extending results to positive characteristic fields.

Contribution

It establishes the openness of the set of K-generic elements in Zariski-dense subgroups and extends the existence results to positive characteristic fields.

Findings

The set of K-generic elements is open in the profinite topology.

Existence of generic elements is confirmed in positive characteristic fields.

Results generalize previous characteristic zero cases.

Abstract

Let G be a semi-simple algebraic group over a finitely generated field K of characteristic zero, and let \Gamma < G(K) be a finitely generated Zariski-dense subgroup. In this note we prove that the set of K-generic elements of \Gamma (whose existence was established earlier in [9]) 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.