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 NG(Γ). We then investigate analogous questions for the automorphism group Aut(G) proving, under appropriate conditions, that StabAut(G)(Γ) is discrete. Finally we show, under appropriate conditions, that the subgroup, of Aut(G) has finite index in StabAut(G)(Γ). We test the limits of our results with various examples and counterexamples.
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