Superrigidity for irreducible lattices and geometric splitting

Monod, Nicolas

doi:10.1090/S0894-0347-06-00525-X

Superrigidity for irreducible lattices and geometric splitting
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by Nicolas Monod

J. Amer. Math. Soc. 19 (2006), 781-814

DOI: https://doi.org/10.1090/S0894-0347-06-00525-X

Published electronically: March 21, 2006

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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

Norbert A’Campo and Marc Burger, Réseaux arithmétiques et commensurateur d’après G. A. Margulis, Invent. Math. 116 (1994), no. 1-3, 1–25 (French). MR 1253187, DOI 10.1007/BF01231555
Michael T. Anderson, The Dirichlet problem at infinity for manifolds of negative curvature, J. Differential Geom. 18 (1983), no. 4, 701–721 (1984). MR 730923
Werner Ballmann, Lectures on spaces of nonpositive curvature, DMV Seminar, vol. 25, Birkhäuser Verlag, Basel, 1995. With an appendix by Misha Brin. MR 1377265, DOI 10.1007/978-3-0348-9240-7
M. B. Bekka, On uniqueness of invariant means, Proc. Amer. Math. Soc. 126 (1998), no. 2, 507–514. MR 1415573, DOI 10.1090/S0002-9939-98-04044-1
Mohammed E. B. Bekka and Alain Valette, Kazhdan’s property $(\textrm {T})$ and amenable representations, Math. Z. 212 (1993), no. 2, 293–299. MR 1202813, DOI 10.1007/BF02571659
Vitaly Bergelson and Joseph Rosenblatt, Mixing actions of groups, Illinois J. Math. 32 (1988), no. 1, 65–80. MR 921351
Philippe Blanc, Sur la cohomologie continue des groupes localement compacts, Ann. Sci. École Norm. Sup. (4) 12 (1979), no. 2, 137–168 (French). MR 543215, DOI 10.24033/asens.1364
Nicolas Bourbaki, Éléments de mathématique. Fasc. II. Livre III: Topologie générale. Chapitre 1: Structures topologiques. Chapitre 2: Structures uniformes, Actualités Scientifiques et Industrielles [Current Scientific and Industrial Topics], No. 1142, Hermann, Paris, 1965 (French). Quatrième édition. MR 0244924
N. Bourbaki, Éléments de mathématique. Première partie. (Fascicule III.) Livre III; Topologie générale. Chap. 3: Groupes topologiques. Chap. 4: Nombres réels, Actualités Sci. Indust., No. 1143, Hermann, Paris, 1960 (French). Troisième édition revue et augmentée. MR 0140603
N. Bourbaki, Éléments de mathématique. Fascicule XXIX. Livre VI: Intégration. Chapitre 7: Mesure de Haar. Chapitre 8: Convolution et représentations, Actualités Scientifiques et Industrielles [Current Scientific and Industrial Topics], No. 1306, Hermann, Paris, 1963 (French). MR 0179291
Martin R. Bridson and André Haefliger, Metric spaces of non-positive curvature, Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 319, Springer-Verlag, Berlin, 1999. MR 1744486, DOI 10.1007/978-3-662-12494-9
Marc Burger, Rigidity properties of group actions on $\textrm {CAT}(0)$-spaces, Proceedings of the International Congress of Mathematicians, Vol. 1, 2 (Zürich, 1994) Birkhäuser, Basel, 1995, pp. 761–769. MR 1403976
Marc Burger, Alessandra Iozzi, and Nicolas Monod, Equivariant embeddings of trees into hyperbolic spaces, Int. Math. Res. Not. 22 (2005), 1331–1369. MR 2152540, DOI 10.1155/IMRN.2005.1331
Jan Dymara and Tadeusz Januszkiewicz, Cohomology of buildings and their automorphism groups, Invent. Math. 150 (2002), no. 3, 579–627. MR 1946553, DOI 10.1007/s00222-002-0242-y
Patrick Eberlein, Isometry groups of simply connected manifolds of nonpositive curvature. II, Acta Math. 149 (1982), no. 1-2, 41–69. MR 674166, DOI 10.1007/BF02392349
É. Ghys and P. de la Harpe (eds.), Sur les groupes hyperboliques d’après Mikhael Gromov, Progress in Mathematics, vol. 83, Birkhäuser Boston, Inc., Boston, MA, 1990 (French). Papers from the Swiss Seminar on Hyperbolic Groups held in Bern, 1988. MR 1086648, DOI 10.1007/978-1-4684-9167-8
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 281122, DOI 10.1090/S0002-9904-1971-12747-7
M. Gromov and P. Pansu, Rigidity of lattices: an introduction, Geometric topology: recent developments (Montecatini Terme, 1990) Lecture Notes in Math., vol. 1504, Springer, Berlin, 1991, pp. 39–137. MR 1168043, DOI 10.1007/BFb0094289
Alain Guichardet, Sur la cohomologie des groupes topologiques. II, Bull. Sci. Math. (2) 96 (1972), 305–332 (French). MR 340464
Jürgen Jost, Nonpositive curvature: geometric and analytic aspects, Lectures in Mathematics ETH Zürich, Birkhäuser Verlag, Basel, 1997. MR 1451625, DOI 10.1007/978-3-0348-8918-6
Jürgen Jost and Shing-Tung Yau, Applications of quasilinear PDE to algebraic geometry and arithmetic lattices, Algebraic geometry and related topics (Inchon, 1992) Conf. Proc. Lecture Notes Algebraic Geom., I, Int. Press, Cambridge, MA, 1993, pp. 169–193. MR 1285380
Jürgen Jost and Shing-Tung Yau, Harmonic maps and superrigidity, Tsing Hua lectures on geometry & analysis (Hsinchu, 1990–1991) Int. Press, Cambridge, MA, 1997, pp. 213–246. MR 1482040
Jürgen Jost and Shing-Tung Yau, Harmonic maps and rigidity theorems for spaces of nonpositive curvature, Comm. Anal. Geom. 7 (1999), no. 4, 681–694. MR 1714940, DOI 10.4310/CAG.1999.v7.n4.a1
Shizuo Kakutani and Kunihiko Kodaira, Über das Haarsche Mass in der lokal bikompakten Gruppe, Proc. Imp. Acad. Tokyo 20 (1944), 444–450 (German). MR 14401
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, DOI 10.4310/CAG.1993.v1.n4.a4
H. Blaine Lawson Jr. and Shing Tung Yau, Compact manifolds of nonpositive curvature, J. Differential Geometry 7 (1972), 211–228. MR 334083
G. A. Margulis, Discrete subgroups of semisimple Lie groups, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], vol. 17, Springer-Verlag, Berlin, 1991. MR 1090825, DOI 10.1007/978-3-642-51445-6

Superrigidity for commensurability subgroups and generalized harmonic maps

Ngaiming Mok, Yum Tong Siu, and Sai-Kee Yeung, Geometric superrigidity, Invent. Math. 113 (1993), no. 1, 57–83. MR 1223224, DOI 10.1007/BF01244302
Nicolas Monod, Arithmeticity vs. nonlinearity for irreducible lattices, Geom. Dedicata 112 (2005), 225–237. MR 2163901, DOI 10.1007/s10711-004-6162-9
Nicolas Monod and Yehuda Shalom, Cocycle superrigidity and bounded cohomology for negatively curved spaces, J. Differential Geom. 67 (2004), no. 3, 395–455. MR 2153026
Nicolas Monod and Yehuda Shalom, Negative curvature from a cohomological viewpoint and cocycle superrigidity, C. R. Math. Acad. Sci. Paris 337 (2003), no. 10, 635–638 (English, with English and French summaries). MR 2030102, DOI 10.1016/j.crma.2003.10.002
Bertrand Rémy, Groupes de Kac-Moody déployés et presque déployés, Astérisque 277 (2002), viii+348 (French, with English and French summaries). MR 1909671

Kac-Moody groups as discrete groups

Bertrand Rémy, Integrability of induction cocycles for Kac-Moody groups, Math. Ann. 333 (2005), no. 1, 29–43. MR 2169827, DOI 10.1007/s00208-005-0663-1
Ju. G. Rešetnjak, Non-expansive maps in a space of curvature no greater than $K$, Sibirsk. Mat. Ž. 9 (1968), 918–927 (Russian). MR 0244922
Viktor Schroeder, A splitting theorem for spaces of nonpositive curvature, Invent. Math. 79 (1985), no. 2, 323–327. MR 778131, DOI 10.1007/BF01388977
Yehuda Shalom, Rigidity of commensurators and irreducible lattices, Invent. Math. 141 (2000), no. 1, 1–54. MR 1767270, DOI 10.1007/s002220000064
J. Tits, Représentations linéaires irréductibles d’un groupe réductif sur un corps quelconque, J. Reine Angew. Math. 247 (1971), 196–220 (French). MR 277536, DOI 10.1515/crll.1971.247.196
T. 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 936083, DOI 10.1007/BF01404454
Robert J. Zimmer, Ergodic theory and semisimple groups, Monographs in Mathematics, vol. 81, Birkhäuser Verlag, Basel, 1984. MR 776417, DOI 10.1007/978-1-4684-9488-4