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


Authors: G. Prasad and A. S. Rapinchuk
Original publication: Trudy Moskovskogo Matematicheskogo Obshchestva, tom 78 (2017), vypusk 2.
Journal: Trans. Moscow Math. Soc. 2017, 299-314
MSC (2010): Primary 20G15, 22E20
DOI: https://doi.org/10.1090/mosc/274
Published electronically: December 1, 2017
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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.


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Additional Information

G. Prasad
Affiliation: Department of Mathematics, University of Michigan, Michigan
Email: gprasad@umich.edu

A. S. Rapinchuk
Affiliation: Department of Mathematics, University of Virginia, Virginia
Email: asr3x@virginia.edu

DOI: https://doi.org/10.1090/mosc/274
Keywords: Zariski-dense subgroups, generic elements, profinite topology
Published electronically: December 1, 2017
Dedicated: Dedicated to E.B.Vinberg on his 80th birthday
Article copyright: © Copyright 2017 G.Prasad, A.S.Rapinchuk

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