On the splitting fields of generic elements in Zariski dense subgroups

Abstract

Let G be a connected, absolutely almost simple, algebraic group defined over a finitely generated, infinite field K, and let Γ be a Zariski dense subgroup of G(K). We show, apart from some few exceptions, that the commensurability class of the field F given by the compositum of the splitting fields of characteristic polynomials of generic elements of Γ determines the group G up to isogeny over the algebraic closure of K.