Finiteness results for lattices in certain Lie groups

Frederick P. Greenleaf Courant Institute of Mathematical Sciences, New York University Martin Moskowitz Ph.D. Program in Mathematics Graduate Center, City University of New York

TBD mathscidoc:1701.333172

Arkiv for Matematik, 48, (2), 311-321, 2008.6
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 $\tilde{\Gamma}=\{i_{\gamma}:\gamma \in \Gamma \},\ i_{\gamma}(x)=\gamma x\gamma^{-1}$ , of Aut($G$) has finite index in Stab_{Aut($G$)}(Γ). We test the limits of our results with various examples and counterexamples.
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  title={Finiteness results for lattices in certain Lie groups},
  author={Frederick P. Greenleaf, and Martin Moskowitz},
  booktitle={Arkiv for Matematik},
Frederick P. Greenleaf, and Martin Moskowitz. Finiteness results for lattices in certain Lie groups. 2008. Vol. 48. In Arkiv for Matematik. pp.311-321.
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