If H is a simple real algebraic subgroup of real rank at least two in a simple real algebraic group G, we prove, in a substantial number of cases, that a Zariski dense discrete subgroup of G containing a lattice in H is a lattice in G. For example, we show that any Zariski dense discrete subgroup of SL n (ℝ) (n≥4) which contains SL 3 (ℤ) (in the top left hand corner) is commensurable with a conjugate of SL n (ℤ). In contrast, when the groups G and H are of real rank one, there are lattices Δ in a real rank one group H embedded in a larger real rank one group G and which extends to a Zariski dense discrete subgroup Γ of G of infinite co-volume .
Chatterji et al. (Wed,) studied this question.