Discrete subgroups of G x G intersecting the diagonal in lattices

Let G be a noncompact simple real Lie group (e.g., G = SL(2,R)), and let Γ be a Zariski dense discrete subgroup of G × G whose intersection with the diagonal subgroup Δ(G) is a lattice. In this talk, we will present some structural constraints on such subgroups Γ and discuss related questions. This is based on ongoing joint work with Sebastian Hurtado.