On Discrete Subgroups Containing a Lattice in a Horospherical Subgroup

We showed in [Oh] that for a simple real Lie group G with real rank at least 2, if a discrete subgroup Γ of G contains lattices in two opposite horospherical subgroups, then Γ must be a non-uniform arithmetic lattice in G, under some additional assumptions on the horospherical subgroups. Somewhat surprisingly, a similar result is true even if we only assume that Γ contains a lattice in one horospherical subgroup, provided Γ is Zariski dense in G.