This article is cited in 5 papers

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

G. Prasad^a, A. S. Rapinchuk<sup>b

a</sup> Department of Mathematics, University of Michigan, USA
^b Department of Mathematics, University of Virginia, USA

Abstract: Let $G$ be a semi-simple 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 note we prove that the set of $K$-generic elements of $\Gamma$ (whose existence was established earlier in [PR301]) 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.

Key words and phrases: Zariski-dense subgroups, generic elements, profinite topology.

UDC: 512.743

MSC: 20G15, 22E20

Received: 30.04.2017
Revised: 09.07.2017

Language: English

 English version: Transactions of the Moscow Mathematical Society, 2017, 78, 299–314

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