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.

**[B]**Kenneth S. Brown,*Buildings*, Springer-Verlag, New York, 1989. MR**969123****[CC]**J. W. Cannon and Daryl Cooper,*A characterization of cocompact hyperbolic and finite-volume hyperbolic groups in dimension three*, Trans. Amer. Math. Soc.**330**(1992), no. 1, 419–431. MR**1036000**, 10.1090/S0002-9947-1992-1036000-0**[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*, Princeton University Press, Princeton, N.J.; University of Tokyo Press, Tokyo, 1973. Annals of Mathematics Studies, No. 78. MR**0385004****[Mu]**J.R. Munkries,*Elements of Algebraic Topology*, Benjamin/Cummings Publishing Company, Menlo Park, 1984.**[Pa]**Pierre Pansu,*Métriques de Carnot-Carathéodory et quasiisométries des espaces symétriques de rang un*, Ann. of Math. (2)**129**(1989), no. 1, 1–60 (French, with English summary). MR**979599**, 10.2307/1971484**[S1]**Richard Evan Schwartz,*The quasi-isometry classification of rank one lattices*, Inst. Hautes Études Sci. Publ. Math.**82**(1995), 133–168 (1996). MR**1383215****[S2]**Richard Evan Schwartz,*Quasi-isometric rigidity and Diophantine approximation*, Acta Math.**177**(1996), no. 1, 75–112. MR**1417087**, 10.1007/BF02392599**[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]**Jacques Tits,*Buildings of spherical type and finite BN-pairs*, Lecture Notes in Mathematics, Vol. 386, Springer-Verlag, Berlin-New York, 1974. MR**0470099****[Zim]**Robert J. Zimmer,*Ergodic theory and semisimple groups*, Monographs in Mathematics, vol. 81, Birkhäuser Verlag, Basel, 1984. MR**776417**

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

Email:
eskin@math.uchicago.edu

DOI:
http://dx.doi.org/10.1090/S0894-0347-98-00256-2

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