Finiteness results for lattices in certain Lie groups

In this note we establish some general finiteness results concerning lattices G 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 Gamma always has finite index in its normalizer N(G)(Gamma). We then investigate analogous questions for the automorphism group Aut(G) proving, under appropriate conditions, that Stab(Aut(G))(Gamma) is discrete. Finally we show, under appropriate conditions, that the subgroup (Gamma) over tilde={i(gamma):gamma is an element of Gamma}, i(gamma)(x)=gamma x gamma(-1), of Aut(G) has finite index in Stab(Aut(G))(Gamma). We test the limits of our results with various examples and counterexamples.