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

Nguyêñ Quôć Thǎńg

J. Lie Theory 13, No. 1, 13-20 (2003).

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:

22E46

Semisimple Lie groups and their representations

Keywords:

semisimple groups; Zariski dense subgroups; \(p\)-adic closures