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)



Superrigidity for irreducible lattices and geometric splitting

Author: Nicolas Monod
Journal: J. Amer. Math. Soc. 19 (2006), 781-814
MSC (2000): Primary 22Exx; Secondary 53Cxx, 20F65
Published electronically: March 21, 2006
MathSciNet review: 2219304
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: We prove general superrigidity results for actions of irreducible lattices on CAT$ (0)$ spaces, first in terms of the ideal boundary, and then for the intrinsic geometry (also for infinite-dimensional spaces). In particular, one obtains a new and self-contained proof of Margulis' superrigidity theorem for uniform irreducible lattices in non-simple groups. The proofs rely on simple geometric arguments, including a splitting theorem which can be viewed as an infinite-dimensional (and singular) generalization of the Lawson-Yau/Gromoll-Wolf theorem. Appendix A gives a very elementary proof of commensurator superrigidity; Appendix B proves that all our results also hold for certain non-uniform lattices.

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

  • [AB] Norbert A'Campo and Marc Burger, Réseaux arithmétiques et commensurateur d'après G. A. Margulis, Invent. Math. 116 (1994), no. 1-3, 1-25. MR 1253187 (96a:22019)
  • [A] Michael T. Anderson, The Dirichlet problem at infinity for manifolds of negative curvature, J. Differential Geom. 18 (1983), no. 4, 701-721 (1984).MR 0730923 (85m:58178)
  • [Ba] Werner Ballmann, Lectures on spaces of nonpositive curvature, DMV Seminar, vol. 25, Birkhäuser Verlag, Basel, 1995, with an appendix by Misha Brin.MR 1377265 (97a:53053)
  • [Be] Mohammed El Bachir Bekka, On uniqueness of invariant means, Proc. Amer. Math. Soc. 126 (1998), no. 2, 507-514. MR 1415573 (98d:43002)
  • [BV] Mohammed El Bachir Bekka and Alain Valette, Kazhdan's property (T) and amenable representations, Math. Z. 212 (1993), no. 2, 293-299. MR 1202813 (94a:22006)
  • [BR] Vitaly Bergelson and Joseph Rosenblatt, Mixing actions of groups, Illinois J. Math. 32 (1988), no. 1, 65-80. MR 0921351 (89g:28029)
  • [Bl] Philippe Blanc, Sur la cohomologie continue des groupes localement compacts, Ann. Sci. École Norm. Sup. (4) 12 (1979), no. 2, 137-168. MR 0543215 (80k:22009)
  • [Bou1] Nicolas Bourbaki, Éléments de mathématique. Fascicule II. Livre III, Topologie générale, Chapitres 1 et 2. Quatrième édition. Actualités Scientifiques et Industrielles, No 1142, Hermann, Paris, 1965. MR 0244924 (39:6237)
  • [Bou2] -, Éléments de mathématique. Fascicule III. Livre III, Topologie générale, Chapitres 3 et 4. Troisième édition. Actualités Scientifiques et Industrielles, No 1143, Hermann, Paris, 1960. MR 0140603 (25:4021)
  • [Bou3] -, Éléments de mathématique, Fascicule XXIX. Livre IV, Intégration, Chapitres 7 et 8, Actualités Scientifiques et Industrielles, No 1306, Hermann, Paris, 1963. MR 0179291 (31:3539)
  • [BH] Martin R. Bridson and André Haefliger, Metric spaces of non-positive curvature, Grundlehren der Mathematischen Wissenschaften 319, Springer, Berlin, 1999.MR 1744486 (2000k:53038)
  • [Bu] Marc Burger, Rigidity properties of group actions on cat$ (0)$-spaces, Proceedings of the International Congress of Mathematicians, Vols. 1, 2 (Zürich, 1994) (Basel), Birkhäuser, 1995, pp. 761-769.MR 1403976 (97j:20033)
  • [BIM] Marc Burger, Alessandra Iozzi, and Nicolas Monod, Equivariant embeddings of trees into hyperbolic spaces, Int. Math. Res. Not. (2005), no. 22, 1331-1369.MR 2152540
  • [DJ] Jan Dymara and Tadeusz Januszkiewicz, Cohomology of buildings and their automorphism groups, Invent. Math. 150 (2002), no. 3, 579-627. MR 1946553 (2003j:20052)
  • [E] Patrick Eberlein, Isometry groups of simply connected manifolds of nonpositive curvature. II, Acta Math. 149 (1982), no. 1-2, 41-69. MR 0674166 (83m:53055)
  • [GH] Étienne Ghys and Pierre de la Harpe (eds.), Sur les groupes hyperboliques d'après Mikhael Gromov, Birkhäuser Verlag, Basel, 1990, Papers from the Swiss Seminar on Hyperbolic Groups held in Bern, 1988. MR 1086648 (92f:53050)
  • [GW] Detlef Gromoll and Joseph A. Wolf, Some relations between the metric structure and the algebraic structure of the fundamental group in manifolds of nonpositive curvature, Bull. Amer. Math. Soc. 77 (1971), 545-552. MR 0281122 (43:6841)
  • [GP] Mikhaïl Gromov and Pierre Pansu, Rigidity of lattices: An introduction, Geometric topology: Recent developments (Montecatini Terme, 1990) (Berlin), Lecture Notes in Math., vol. 1504, Springer, Berlin, 1991, pp. 39-137.MR 1168043 (93f:53036)
  • [G] Alain Guichardet, Sur la cohomologie des groupes topologiques, II, Bull. Sci. Math. (2) 96 (1972), 305-332. MR 0340464 (49:5219)
  • [J] Jürgen Jost, Nonpositive curvature: Geometric and analytic aspects, Lectures in Mathematics ETH Zürich, Birkhäuser Verlag, Basel, 1997. MR 1451625 (98g:53070)
  • [JY1] Jürgen Jost and Shing-Tung Yau, Applications of quasilinear PDE to algebraic geometry and arithmetic lattices, Conf. Proc. Lecture Notes Algebraic Geom., I, Internat. Press, Cambridge, MA, 1993, pp. 169-193. MR 1285380 (95g:32029)
  • [JY2] -, Harmonic maps and superrigidity, Tsing Hua lectures on geometry & analysis (Hsinchu, 1990-1991), Internat. Press, Cambridge, MA, 1997, pp. 213-246. MR 1482040 (99i:53052)
  • [JY3] -, Harmonic maps and rigidity theorems for spaces of nonpositive curvature, Comm. Anal. Geom. 7 (1999), no. 4, 681-694.MR 1714940 (2000m:53055)
  • [KK] Shizuo Kakutani and Kunihiko Kodaira, Über das Haarsche Maßin der lokal bikompakten Gruppe, Proc. Imp. Acad. Tokyo 20 (1944), 444-450.MR 0014401 (7:279d)
  • [KS] Nicholas J. Korevaar and Richard M. Schoen, Sobolev spaces and harmonic maps for metric space targets, Comm. Anal. Geom. 1 (1993), no. 3-4, 561-659.MR 1266480 (95b:58043)
  • [LY] H. Blaine Lawson, Jr. and Shing-Tung Yau, Compact manifolds of nonpositive curvature, J. Differential Geometry 7 (1972), 211-228. MR 0334083 (48:12402)
  • [M1] Grigori A. Margulis, Discrete subgroups of semisimple Lie groups, Springer-Verlag, Berlin, 1991. MR 1090825 (92h:22021)
  • [M2] -, Superrigidity for commensurability subgroups and generalized harmonic maps, unpublished manuscript.
  • [MSY] Ngaiming Mok, Yum Tong Siu, and Sai-Kee Yeung, Geometric superrigidity, Invent. Math. 113 (1993), no. 1, 57-83. MR 1223224 (94h:53079)
  • [Md] Nicolas Monod, Arithmeticity vs. nonlinearity for irreducible lattices, Geom. Dedicata 112 (2005), 225-237. MR 2163901
  • [MS1] Nicolas Monod and Yehuda Shalom, Cocycle superrigidity and bounded cohomology for negatively curved spaces, J. Differential Geometry 67 (2004), 395-455. MR 2153026
  • [MS2] -, Negative curvature from a cohomological viewpoint and cocycle superrigidity, C. R. Acad. Sci. Paris Sér. I Math. 337 (2003), no. 10, 635-638. MR 2030102 (2004j:20091)
  • [R*] Bertrand Rémy, Groupes de Kac-Moody déployés et presque déployés, Astérisque (2002), no. 277, viii+348. MR 1909671 (2003d:20036)
  • [R1] -, Kac-Moody groups as discrete groups, Preprint, 2003.
  • [R2] -, Integrability of induction cocycles for Kac-Moody groups, Math. Ann. 333 (2005), 29-43. MR 2169827
  • [Rk] Yuri{\u{\i\/}}\kern.15em G. Reshetnyak, Non-expansive maps in a space of curvature no greater than $ K$, Sibirsk. Mat. Z. 9 (1968), 918-927. MR 0244922 (39:6235)
  • [Sch] Viktor Schroeder, A splitting theorem for spaces of nonpositive curvature, Invent. Math. 79 (1985), no. 2, 323-327. MR 0778131 (86b:53041)
  • [Sh] Yehuda Shalom, Rigidity of commensurators and irreducible lattices, Invent. Math. 141 (2000), no. 1, 1-54. MR 1767270 (2001k:22022)
  • [T] Jacques Tits, Représentations linéaires irréductibles d'un groupe réductif sur un corps quelconque, J. Reine Angew. Math. 247 (1971), 196-220.MR 0277536 (43:3269)
  • [V] Tyakal N. Venkataramana, On superrigidity and arithmeticity of lattices in semisimple groups over local fields of arbitrary characteristic, Invent. Math. 92 (1988), no. 2, 255-306. MR 0936083 (89c:20073)
  • [Z] Robert J. Zimmer, Ergodic theory and semisimple groups, Birkhäuser Verlag, Basel, 1984.MR 0776417 (86j:22014)

Similar Articles

Retrieve articles in Journal of the American Mathematical Society with MSC (2000): 22Exx, 53Cxx, 20F65

Retrieve articles in all journals with MSC (2000): 22Exx, 53Cxx, 20F65

Additional Information

Nicolas Monod
Affiliation: Department of Mathematics, The University of Chicago, 5734 University Avenue, Chicago, Illinois 60637
Address at time of publication: Université de Genève, 2-4, rue du Lièvre, CP 64, CH-1211 Genève 4, Switzerland

Keywords: Superrigidity, splitting, lattices, Hadamard spaces.
Received by editor(s): December 13, 2004
Published electronically: March 21, 2006
Article copyright: © Copyright 2006 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

American Mathematical Society