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On isomorphism classes of Zariski dense subgroups of semisimple algebraic groups with isomorphic \(p\)-adic closures. (English) Zbl 1012.22020

The author states a result saying that there is only a finite number of isomorphism classes of groups as stated in the title assuming that the groups involved are “big” which basically means that they are closed in the congruence topology. The idea of proof is taken from papers of Grunewald, Pickel and Segal, in particular [F. J. Grunewald, P. F. Pickel and D. Segal, Ann. Math. (2) 111, 155–195 (1980; Zbl 0431.20033)]. The author gives an outline of proof.

MSC:

22E40 Discrete subgroups of Lie groups
20G25 Linear algebraic groups over local fields and their integers
22E35 Analysis on \(p\)-adic Lie groups
22E45 Representations of Lie and linear algebraic groups over real fields: analytic methods

Citations:

Zbl 0431.20033
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References:

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