Remote Access Journal of the American Mathematical Society
Green Open Access

Journal of the American Mathematical Society

ISSN 1088-6834(online) ISSN 0894-0347(print)



Quasi-isometric rigidity of nonuniform lattices
in higher rank symmetric spaces

Author: Alex Eskin
Journal: J. Amer. Math. Soc. 11 (1998), 321-361
MSC (1991): Primary 22E40, 20F32
MathSciNet review: 1475886
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: We compute the quasi-isometry group of an irreducible nonuniform lattice in a semisimple Lie group with finite center and no rank one factors, and show that any two such lattices are quasi-isometric if and only if they are commensurable up to conjugation.

References [Enhancements On Off] (What's this?)

  • [B] K. Brown, Buildings, Springer-Verlag, New York, 1989. MR 90e:20001
  • [CC] J. Cannon and D. Cooper, A characterization of cocompact hyperbolic and finite-volume hyperbolic groups in dimension three, Trans. AMS 330 (1992), 419-431. MR 92f:22017
  • [Dru] C. Drutu, Quasi-isometric classification of semisimple groups in higher rank, Preprint.
  • [EF] A. Eskin and B. Farb, Quasi-flats and rigidity in higher rank symmetric spaces, Journal Amer. Math. Soc. Vol 10, No. 3, 1997, pp. 653-692. CMP 97:11
  • [Fa] B. Farb, The quasi-isometry classification of lattices in semisimple Lie groups, Math. Res. Lett. Vol. 4, No. 5 (1997), pp. 705-717.
  • [FM] B. Farb and L. Mosher, A rigidity theorem for the solvable Baumslag-Solitar groups (with an appendix by D. Cooper), to appear in Inventiones Math.
  • [FS] B. Farb and R. Schwartz, The large-scale geometry of Hilbert modular groups, J. Diff. Geom. 44, No.3 (1996), pp 435-478. CMP 97:07
  • [KL] B. Kleiner and B. Leeb, Rigidity of quasi-isometries for symmetric spaces of higher rank, to appear in Publ. Math. IHES.
  • [LMR] A. Lubotzky, S. Mozes, M. S. Raghunathan, The Word and Riemannian Metrics on Lattices of Semisimple Groups, preprint.
  • [Mo] G.D. Mostow, Strong Rigidity of Locally Symmetric Spaces, Annals of Math. Studies, No. 78, Princeton Univ. Press, 1973. MR 52:5874
  • [Mu] J.R. Munkries, Elements of Algebraic Topology, Benjamin/Cummings Publishing Company, Menlo Park, 1984.
  • [Pa] P. Pansu, Metriques de Carnot-Caratheodory et quasiisometries des espaces symmetriques de rang un, Annals of Math. 129 (1989), p.1-60. MR 90e:53058
  • [S1] R. Schwartz, The Quasi-Isometry Classification of Rank One Lattices, IHES Sci. Publ. Math., vol. 82, (1996). MR 97c:22014
  • [S2] R. Schwartz, Quasi-Isometric Rigidity and Diophantine Approximation, Acta Math. 177 (1996), No. 1, pp. 75-112. MR 97m:53093
  • [Sh] N. Shah, Invariant measures and orbit closures on homogeneous spaces for actions of subgroups generated by unipotent elements, To appear in the Proceedings of the International Colloquium on Lie Groups and Ergodic Theory, TIFR, Bombay, 1996.
  • [Ti] J. Tits, Buildings of spherical type and finite BN-pairs, Lecture Notes in Math., Vol. 386, Springer-Verlag, 1974. MR 57:9866
  • [Zim] R. Zimmer, Ergodic Theory and Semisimple Groups, Birkhauser Boston, Inc., 1984. MR 86j:22014

Similar Articles

Retrieve articles in Journal of the American Mathematical Society with MSC (1991): 22E40, 20F32

Retrieve articles in all journals with MSC (1991): 22E40, 20F32

Additional Information

Alex Eskin
Affiliation: Department of Mathematics, University of Chicago, 5734 S.University Ave, Chicago, Illinois 60637

Keywords: Lie groups, discrete subgroups, geometric group theory
Received by editor(s): October 28, 1996
Received by editor(s) in revised form: October 21, 1997
Additional Notes: The author was supported in part by an N.S.F. Postdoctoral Fellowship.
Article copyright: © Copyright 1998 American Mathematical Society

American Mathematical Society