Geodesic
Generic elements of a~Zariski-dense subgroup form an open subset
Trudy Moskovskogo matematičeskogo obŝestva, Trudy Moskovskogo Matematicheskogo Obshchestva, Tome 78 (2017) no. 2, pp. 357-375.

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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. 
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G. Prasad; A. S. Rapinchuk. Generic elements of a~Zariski-dense subgroup form an open subset. Trudy Moskovskogo matematičeskogo obŝestva, Trudy Moskovskogo Matematicheskogo Obshchestva, Tome 78 (2017) no. 2, pp. 357-375. http://geodesic.mathdoc.fr/item/MMO_2017_78_2_a7/

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