Quasiisometric rigidity of nonuniform lattices in higher rank symmetric spaces
Author:
Alex Eskin
Journal:
J. Amer. Math. Soc. 11 (1998), 321361
MSC (1991):
Primary 22E40, 20F32
MathSciNet review:
1475886
Fulltext PDF Free Access
Abstract 
References 
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Abstract: We compute the quasiisometry 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 quasiisometric if and only if they are commensurable up to conjugation.
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 K. Brown, Buildings, SpringerVerlag, New York, 1989. MR 90e:20001
 [CC]
 J. Cannon and D. Cooper, A characterization of cocompact hyperbolic and finitevolume hyperbolic groups in dimension three, Trans. AMS 330 (1992), 419431. MR 92f:22017
 [Dru]
 C. Drutu, Quasiisometric classification of semisimple groups in higher rank, Preprint.
 [EF]
 A. Eskin and B. Farb, Quasiflats and rigidity in higher rank symmetric spaces, Journal Amer. Math. Soc. Vol 10, No. 3, 1997, pp. 653692. CMP 97:11
 [Fa]
 B. Farb, The quasiisometry classification of lattices in semisimple Lie groups, Math. Res. Lett. Vol. 4, No. 5 (1997), pp. 705717.
 [FM]
 B. Farb and L. Mosher, A rigidity theorem for the solvable BaumslagSolitar groups (with an appendix by D. Cooper), to appear in Inventiones Math.
 [FS]
 B. Farb and R. Schwartz, The largescale geometry of Hilbert modular groups, J. Diff. Geom. 44, No.3 (1996), pp 435478. CMP 97:07
 [KL]
 B. Kleiner and B. Leeb, Rigidity of quasiisometries 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 CarnotCaratheodory et quasiisometries des espaces symmetriques de rang un, Annals of Math. 129 (1989), p.160. MR 90e:53058
 [S1]
 R. Schwartz, The QuasiIsometry Classification of Rank One Lattices, IHES Sci. Publ. Math., vol. 82, (1996). MR 97c:22014
 [S2]
 R. Schwartz, QuasiIsometric Rigidity and Diophantine Approximation, Acta Math. 177 (1996), No. 1, pp. 75112. 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 BNpairs, Lecture Notes in Math., Vol. 386, SpringerVerlag, 1974. MR 57:9866
 [Zim]
 R. Zimmer, Ergodic Theory and Semisimple Groups, Birkhauser Boston, Inc., 1984. MR 86j:22014
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Additional Information
Alex Eskin
Affiliation:
Department of Mathematics, University of Chicago, 5734 S.University Ave, Chicago, Illinois 60637
Email:
eskin@math.uchicago.edu
DOI:
http://dx.doi.org/10.1090/S0894034798002562
PII:
S 08940347(98)002562
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
