Arkiv för Matematik

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

Finiteness results for lattices in certain Lie groups

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
First available in Project Euclid: 31 January 2017

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

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

References

• Abbaspour, H. and Moskowitz, M., Basic Lie Theory, World Scientific, Singapore, 2007.
• Barbano, P., Automorphisms and quasi conformal mappings of Heisenberg type groups, J. Lie Theory 8 (1998), 255–277.
• Crandall, G. and Dodziuk, J., Integral structures on H-type Lie algebras, J. Lie Theory 12 (2002), 69–79.
• Greenleaf, F. and Moskowitz, M., Lattices in certain two-step solvable Lie groups of solvable Lie groups: Estimates on StabAut(G)(Γ)/Γ, In preparation.
• Greenleaf, F., Moskowitz, M. and Rothschild, L., Unbounded conjugacy classes in Lie groups and the location of central measures, Acta Math. 132 (1974), 225–243.
• Greenleaf, F., Moskowitz, M. and Rothschild, L., Automorphisms, orbits and homogeneous spaces of non-connected Lie groups, Math. Ann. 212 (1974), 145–155.
• Mosak, R. and Moskowitz, M., Zariski density in Lie groups, Israel J. Math. 52 (1985), 1–14.
• Mosak, R. and Moskowitz, M., Stabilizers of lattices in Lie groups, J. Lie Theory 4 (1994), 1–16.
• Mosak, R. and Moskowitz, M., Lattices in a split solvable Lie group, Math. Proc. Cambridge Philos. Soc. 122 (1997), 245–250.
• Moskowitz, M., On the density theorems of Borel and Furstenberg, Ark. Mat. 16 (1978), 11–27.
• Moskowitz, M., Some results on automorphisms of bounded displacement and bounded cocycles, Monatsh. Math. 85 (1978), 323–336.
• Moskowitz, M., An extension of Mahler’s theorem to simply connected nilpotent groups, Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. Rend. Lincei Mat. Appl. 16 (2005), 265–270.
• Raghunathan, M. S., Discrete Subgroups of Lie Groups, Springer, Berlin, 1972.
• Whitney, H., Elementary structure of real algebraic varieties, Ann. of Math. 66 (1957), 545–556.