Combinations of Non-Archimedean Discrete Groups

 Chatterji and Venkataramana, motivated by Nori’s problem, asked for a real algebraic semisimple Lie group G and a simple Lie subgroup H of G, whether a Zariski-dense discrete subgroup of G, intersecting H in a lattice, is itself a lattice in G. After discussing what is known related to this question and Nori’s problem, we will address a similar version of this question for the k-points of semisimple groups defined over a non-Archimedean local field k, and provide examples for which a positive answer fails. This is based on a combination theorem for discrete subgroups of matrix groups over discretely valued fields. We will also discuss a construction of non-Fuchsian hyperbolic groups that quasi-isometrically embed as subgroups of matrix groups over a discretely valued field.