Arkiv för Matematik

  • Ark. Mat.
  • Volume 48, Number 2 (2010), 311-321.

Finiteness results for lattices in certain Lie groups

Frederick P. Greenleaf and Martin Moskowitz

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Abstract

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 $\tilde{\Gamma}=\{i_{\gamma}:\gamma \in \Gamma \},\ i_{\gamma}(x)=\gamma x\gamma^{-1}$, of Aut(G) has finite index in StabAut(G)(Γ). We test the limits of our results with various examples and counterexamples.

Dedication

This paper is dedicated to the memory of our colleague Larry Corwin.

Article information

Source
Ark. Mat., Volume 48, Number 2 (2010), 311-321.

Dates
Received: 24 June 2008
First available in Project Euclid: 31 January 2017

Permanent link to this document
https://projecteuclid.org/euclid.afm/1485907115

Digital Object Identifier
doi:10.1007/s11512-009-0112-6

Mathematical Reviews number (MathSciNet)
MR2672612

Zentralblatt MATH identifier
1202.22015

Rights
2009 © Institut Mittag-Leffler

Citation

Greenleaf, Frederick P.; Moskowitz, Martin. Finiteness results for lattices in certain Lie groups. Ark. Mat. 48 (2010), no. 2, 311--321. doi:10.1007/s11512-009-0112-6. https://projecteuclid.org/euclid.afm/1485907115


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