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Superrigidity for homomorphisms into isometry groups of \(\text{CAT}(-1)\) spaces. (English) Zbl 0888.22008
A \(\text{CAT}(-1)\) space is a geodesic metric space in which the comparison maps of its geodesic triangles into the usual hyperbolic plane are distance increasing. In the paper three types of rigidity results related to \(\text{CAT}(-1)\) spaces are proved, namely the rigidity of the actions on \(\text{CAT}(-1)\) spaces under the commensurability subgroups, the higher rank lattices and certain ergodic cocycles.
Reviewer: K.Riives (Tartu)

MSC:
22E40 Discrete subgroups of Lie groups
22D40 Ergodic theory on groups
53A55 Differential invariants (local theory), geometric objects
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[1] [Ad] S. Adams,Reduction of cocycles with hyperbolic targets, preprint.
[2] [Ba] W. Ballmann,Lectures on Spaces of Nonpositive Curvature, DMV, Birkhäuser, 1995.
[3] [Bu] M. Burger,Rigidity properties of group actions on CAT(0)-spaces, Proceedings of the ICM, Zürich, Switzerland (1994), Birkhäuser, 1995, 761-769. · Zbl 0838.53052
[4] [BM] M. Burger, S. Mozes, CAT(?1)spaces, divergence groups and their commensurators, J. Amer. Math. Soc.9 (1996), 57-93. · Zbl 0847.22004 · doi:10.1090/S0894-0347-96-00196-8
[5] [CFS] I. P. Cornfeld, S. V. Fomin, Ya. G. Sinai,Ergodic Theory, Springer-Verlag, 1982.
[6] [F] G. Folland,Real Analysis, John Wiley & Sons, 1984. · Zbl 0549.28001
[7] [GH] E. Ghys, P. de la Harpe (Editors),Sur les Groupes Hyperboliques d’après Mikhael Gromov, Progress in Mathematics, Vol. 83, Birkhäuser, 1990. · Zbl 0731.20025
[8] [GHV] E. Ghys, A. Haefliger, A. Verjovsky (Editors),Group Theory from a Geometrical Viewpoint, World Scientific, 1991. · Zbl 0809.00017
[9] [GLP] M. Gromov,Structures Métriques pour les Variétés Riemanniennes, rédigé par J. Lafontaine et P. Pansu, Cedic/F. Nathan, 1981.
[10] [Gr] M. Gromov,Hyperbolic Groups, Essays in Group Theory, edited by S. M. Gersten, MSRI series, Vol. 8, Springer-Verlag, 1987.
[11] [LMZ] A. Lubotzky, S. Mozes, R. Zimmer,Superrigidity for the commensurability group of tree lattices, Comment. Math. Helv.69 (1994), 523-548. · Zbl 0839.22011 · doi:10.1007/BF02564503
[12] [Ma1] G. A. Margulis,Discrete Subgroups of Semisimple Lie Groups, Springer-Verlag, 1991.
[13] [Ma2] G. A. Margulis,On the decomposition of discrete subgroup into amalgams, Sel. Math. Sov.1 (1981), 197-213. · Zbl 0515.20031
[14] [Ma3] ?/ A. ????????, ??????-????? ????????qj ???????? ? ?????? ???, ????? ?????? ? ??? ????.12, 4 (1978), 64-80. English translation: G. A. Margulis,Quotient groups of discrete subgroups and measure theory, Funct. Anal. Appl.12 (1978), 295-305.
[15] [Ma4] G. A. Margulis,Superrigidity of commensurability subgroups and generalized harmonic maps, 1994.
[16] [MT] G. A. Margulis, G. M. Tomanov,Invariant measures for actions of unipotent groups over local fields on homogeneous spaces, Invent. Math.116 (1994), 347-392. · Zbl 0816.22004 · doi:10.1007/BF01231565
[17] [Mo] G. D. Mostow,Strong Rigidity of Locally Symmetric Spaces, Annals of Math. Studies78, Princeton Univ. Press, 1973. · Zbl 0265.53039
[18] [Sp] R. J. Spatzier,Harmonic analysis in rigidity theory, Ergodic theory and its connections with harmonic analysis (Alexandria, 1993), 153-205, London Math. Soc. Lecture Note Ser., Vol. 205, Cambridge Univ. Press, Cambridge, 1995.
[19] [St] R. Steinberg,Lectures on Chevalley Groups, Yale University, 1967.
[20] [Z] R. J. Zimmer,Ergodic Theory and Semisimple Groups, Birkhäuser, 1984. · Zbl 0571.58015
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