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

2017/07/09 by Prasad, Gopal, Rapinchuk, Andrei S.
#FOS: Mathematics #Group Theory (math.GR) #Number Theory (math.NT)

paper · doi:10.48550/arxiv.1707.02508

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.

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