## Zariski dense subgroups of semisimple algebraic groups with isomorphic $$p$$-adic closures.(English)Zbl 1016.22007

The aim of this paper is to prove under certain natural conditions the finiteness of the number of isomorphism classes of Zariski dense subgroups in semisimple groups with isomorphic $$p$$-adic closures. More concretely, let $$G_i$$ be a Zariski dense subgroup in a simply connected absolutely almost simple $$\mathbf Q$$-group $${\mathbf G}_i$$ $$(i\in I)$$ such that $$G_i\subset {\mathbf G}_i ({\mathbf Z})$$ and $$G_i$$ is “big” in $${\mathbf G}_i$$. Assume that the groups $$G_i$$ are mutually non-isomorphic, while their $$p$$-adic closures are topologically isomorphic for all primes $$p$$. Then the set $$\{G_i\}_{i\in I}$$ is the disjoint union of a finite number of isomorphism classes.
Reviewer: Li Fu-an (Beijing)

### MSC:

 2.2e+47 Semisimple Lie groups and their representations