Finiteness results for lattices in certain Lie groups

In this note we establish some general finiteness results concerning lattices Γ in connected Lie groups G which possess certain "density" properties (see Moskowitz, M., On the density theorems of Borel and Furstenberg, Ark. Mat. 16 (1978), 11-27, and Moskowitz, M., Some results on automorphisms of bounded displacement and bounded cocycles, Monatsh. Math. 85 (1978), 323-336). For such groups we show that Γ always has finite index in its normalizer N_G(Γ). We then investigate analogous questions for the automorphism group Aut(G) proving, under appropriate conditions, that Stab_Aut(G)(Γ) is discrete. Finally we show, under appropriate conditions, that the subgroup, of Aut(G) has finite index in Stab_Aut(G)(Γ). We test the limits of our results with various examples and counterexamples.