Abstract
The aim of this note is to give a geometric proof for classical local rigidity of lattices in semisimple Lie groups. We are reproving well known results in a more geometric (and hopefully clearer) way.
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References
Ballmann, W., Gromov, M., and Schroeder, V.:Manifolds of Nonpositive Curvature, Birkhauser, Basel, 1985.
Besson, G., Courtois, G., and Gallot, S.:Entropies et rigidit*#x00E9;s des espaces localement sym*#x00E9;triques de courbure strictement n*#x00E9;gative, Geom.Funct.Anal. 5(5) (1995), 731–799.
Calabi, E.:On Compact, Riemannian Manifolds with Constant Curvature.I.Proc.Sympos. Pure Math., Amer.Math.Soc., Providence, R.I., 1961, Vol.III, pp.155–180.
Canary, R.D., Epstein, D.B.A., and Green, P.:Notes on notes of Thurston.Analytical and Geometric Aspects of Hyperbolic Space(Coventry/Durham, 1984), London Math. Soc.Lecture Note Ser.111, Cambridge Univ.Press, Cambridge, 1987, pp.3–92.
Ehresmann, C.:Sur les espaces localement homog*#x00E8;nes.Enseign.Math.(1936), 5–6.
Ehresmann, C.:Les connexions in nit*#x00E9;simales dans un br*#x00E9;diff*#x00E9;rentiable.Extrait du’ Colloque de Topologie,'Bruxelles, 1950, CBRM.
Garland, H.and Raghunathan, M.S.:Fundamental domains for lattices in(R-)rank 1 Lie groups, Ann.of Math. 92(1970), 279–326.
Gromov, M.:Asymptotic invariants of in nite groups, Geometric group theory, In: G.A. Niblo and M.A. Roller(eds), Proc.Sympos.in Sussex, 1991, Vol.2, London Math. Soc.Lectures Notes, Cambrige Univ.Press, 1993.
Leuzinger, E.:An exhaustion of locally symmetric spaces by compact submanifolds with corners, Invent.Math. 121(2)(1995), 389–410.
Lubotzky, A., Mozes, S., and Raghunathan, M.S.:The word and Riemannian metrics on lattices of semisimple groups.Inst.Hautes *#x00C9;tudes Sci.Publ.Math. 91(2000), 5–53(2001).
Lubotzky, A., Mozes, S., and Raghunathan, M.S.:Cyclic subgroups of exponential growth and metrics on discrete groups, C.R.Acad.Sci.Paris Ser.I Math. 317(8) (1993), 735–740.
Maĺcev, A.I.:On class of homogeneous spaces, Amer.Math.Soc.Transl. 39(1951).
Margulis, G.A.:Arithmeticity of irreducible lattices of semisimple groups of rank greater then 1, Invent.Math 76(1)(1984), 93–120.
Margulis, G.A.:Non-uniform lattices in semisimple algebraic groups, In:Lie Groups and their Representations(Proc.Summer School Group Representations of the Bolyai Jos Math. Soc., Budapest, 1971), Halsted, New York, 1975, pp.371–553.
Mostow, G.D.:Strong Rigidity of Locally Symmetric Spaces, Ann.of Math.Stud.78, Princeton University Press, Princeton, NJ, University of Tokyo Press, Tokyo, 1973.
Raghunathan, M.S.:Discrete Subgroups of Lie Groups, Springer, New York, 1972.
Selberg, A.:On discontinuous groups in higher-dimension symmetric spaces, In:Contributions to Functional Theory, Tata Institute, Bombay, 1960, pp.147–164.
Thurston, W.P.:The Geometry and Topology of Three-Manifolds, Lectures Notes, Princeton Univ., Princeton, NJ, 1979.
Thurston, W.P.:Three-Dimensional Geometry and Topology, Vol.1, Princeton Univ. Press, 1997.
Tits, J.:Free subgroups in linear groups, J.Algebra 20(1972).
Wang, H.C.:Topics on totally discontinuous groups, In:W. Boothby and G. Weiss(eds), Symmetric Spaces, Marcel Dekker, 1972, pp.460–468.
Weil, A.:On discrete subgroups of Lie groups II, Ann.of Math.75(1962), 578–602.
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Bergeron, N., Gelander, T. A Note on Local Rigidity. Geometriae Dedicata 107, 111–131 (2004). https://doi.org/10.1023/B:GEOM.0000049122.75284.06
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DOI: https://doi.org/10.1023/B:GEOM.0000049122.75284.06