Hostname: page-component-cd9895bd7-q99xh Total loading time: 0 Render date: 2024-12-27T07:21:34.245Z Has data issue: false hasContentIssue false
  • English
  • Français

Deficiencies of lattices in connected Lie groups

Published online by Cambridge University Press:  17 April 2009

Jonathan A. Hillman
Jonathan A. Hillman
Affiliation:
School of Mathematics and Statistics, The University of Sydney, Sydney, NSW 2006, Australia, e-mail: john@maths.usyd.edu.au
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

We complete the determination of the groups of positive deficiency which occur as lattices in connected Lie groups. The torsion free groups among them are 3-mainfold groups. We show that any other torsion free 3-manifold group which is such a lattice is the group of an aspherical closed geometric 3-manifold.

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 65 , Issue 3 , June 2002 , pp. 393 - 397
Copyright
Copyright © Australian Mathematical Society 2002

References

[1]Auslander, L., ‘On radicals of discrete subgroups of Lie groups’, Amer. J. Math. 85 (1963), 145–150.CrossRefGoogle Scholar
[2]Borel, A. and Serre, J.-P., ‘Corners and arithmetic groups’, Comment. Math. Helvetici 48 (1973), 436–489.CrossRefGoogle Scholar
[3]Gabai, D., ‘Convergence groups are Fuchsian groups’, Ann. of Math. 136 (1992), 447–510.CrossRefGoogle Scholar
[4]Gabai, D., Meyerhoff, G.R. and Thurston, N., ‘Homotopy hyperbolic 3-manifolds are hyperbolic’, (preprint, MSRI 1996–058), Ann. of Math. (to appear).Google Scholar
[5]Helgason, S., Differential geometry, Lie groups and symmetric spaces, Graduate Studies in Mathematics 34 (Academic Press Inc., London, New York, 1978).Google Scholar
[6]Katok, S., Fuchsian groups, Chicago Lectures in Mathematics (The University of Chicago Press, Chicago, London, 1992).Google Scholar
[7]Lott, J., ‘Deficiencies of lattice subgroups of Lie groups’, Bull. London Math. Soc. 31 (1999), 191–196.CrossRefGoogle Scholar
[8]Lubotzky, A., ‘Group presentations, p-adic Lie groups and lattices in SL2 (ℂ)’, Ann. of Math. 118 (1983), 115–130.CrossRefGoogle Scholar
[9]Margulis, G.A., Discrete subgroups of semisimple Lie groups, Ergebnisse der Math. 3 Folge Band 17 (Springer-Verlag, Berlin, Heidelberg, New York, 1991).CrossRefGoogle Scholar
[10]Ragunathan, M., Discrete subgroups of Lie groups, Ergebnisse der Mathematik Band 68 (Springer-Verlag, Berlin, Heidelberg, New York, 1972).CrossRefGoogle Scholar
[11]Raymond, F. and Vasquez, A.T., ‘3-Manifolds whose universal coverings are Lie groups’, Topology Appl. 12 (1981), 161–179.CrossRefGoogle Scholar
[12]Thurston, W.P., ‘Three dimensional manifolds, Kleinian groups and hyperbolic geometry’, Bull. Amer. Math. Soc. 6 (1982), 357–381.CrossRefGoogle Scholar
[13]Zimmer, R.J., Ergodic theory and semisimple Lie groups (Birkhäser, Boston, Basel, Stuttgart, 1984).CrossRefGoogle Scholar