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4 - Compact Clifford–Klein Forms – Geometry, Topology and Dynamics

Published online by Cambridge University Press:  05 January 2016

David Constantine
Affiliation:
Wesleyan University, Middletown
C. S. Aravinda
Affiliation:
TIFR Centre for Applicable Mathematics, Bangalore, India
F. T. Farrell
Affiliation:
Tsinghua University, Beijing
J. -F. Lafont
Affiliation:
Ohio State University
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Print publication year: 2016

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References

[1] Abels, H. 2001. Properly discontinuous groups of affine transformations, a survey. Geometriae Dedicata, 87, 309–333.CrossRefGoogle Scholar
[2] Barbot, Thierry. 2013. Deformations of Fuchsian AdS representations are quasi-Fuchsian. arXiv:1301.4309.
[3] Benoist, Yves. 1994. Actions propres de groupes libres sur les espaces homogè nes réductifs. Comptes Rendus de l'Académie des Sciences. Série I. Mathématique Comptes Rendus de l'Académie des Sciences - Series I - Mathematics, 319(9),937–940.Google Scholar
[4] Benoist, Yves. 1996. Actions propres sur les espaces homogènes réductifs. Annals of Mathematics, 144, 315–347.CrossRefGoogle Scholar
[5] Benoist, Yves, and Labourie, François. 1992. Sur les espaces homogènes modèles de variétés compactes. PublicationsMathématiques de l'I.H.É.S., 76, 99–109.Google Scholar
[6] Borel, Armand. 1963. Compact Clifford-Klein forms of symmetric spaces. Topology,2, 111–122.CrossRefGoogle Scholar
[7] Brown, Kenneth S. 1994. Cohomology of groups. Graduate Texts in Mathematics, vol. 87. New York: Springer-Verlag.Google Scholar
[8] Calabi, E., and Markus, L. 1962. Relativistic space forms. Annals of Mathematics, 75, 63–76.CrossRefGoogle Scholar
[9] Charette, Virginie, Drumm, Todd, Goldman, William, and Morrill, Maria. 2003. Complete flat affine and Lorentzian manifolds. Geometriae Dedicata, 97, 187–198.Google Scholar
[10] Constantine, David. 2011. Compact forms of homogeneous spaces and higher-rank semisimple group actions. arXiv:1209.3940.
[11] Ehresmann, C. 1950. Les connexions infinitésimales dans un espace fibré différentiable. Pages 29–55 of: Colloque de topologie (espaces fibrés). Bruxelles.Google Scholar
[12] Ghys, É. 1987. Flots d'Anosov dont les feuilletages stables sont différentiables. Annales Scientifiques de l’École Normale Supérieure, 20(2), 251–270.Google Scholar
[13] Ghys, É. 1995. Déformation des structures complexes sur les espaces homogènes de SL(2,C). Journal für die reine und angewandte Mathematik, 468, 113–138.Google Scholar
[14] Goldman, William M. 1985. Nonstandard Lorentz space forms. Journal of Differential Geometry, 21(2), 301–308.CrossRefGoogle Scholar
[15] Guéritaud, François, and Kassel, Fanny. 2013. Maximally stretched laminations on geometrically finite hyperbolic manifolds. arXiv:1307.0250.
[16] Guichard, Olivier, and Wienhard, Anna. 2012. Anosov Representations: domains of discontinuity and applications. Inventiones Mathematicae, 190(2), 357–438.CrossRefGoogle Scholar
[17] Iozzi, Alessandra, and Witte-Morris, Dave. 2004. Tessellations of homogeneous spaces of classical groups of real rank two. Geometriae Dedicata, 103, 115–191.CrossRefGoogle Scholar
[18] Kassel, Fanny. Quotients compacts d'espaces homogènes réels ou p-adiques. Ph.D.Thesis, Université Paris-Sud 11, November, 2009. (available at http://math.univlille1.fr/~kassel/These.pdf).
[19] Kassel, Fanny. 2008. Proper actions on corank-one reductive homogeneous spaces. Journal of Lie Theory, 18, 961–978.Google Scholar
[20] Kassel, Fanny. 2012. Deformation of proper actions on reductive homogeneous spaces. Mathematische Annalen, 353(2), 599–632.CrossRefGoogle Scholar
[21] Kassel, Fanny, and Kobayashi, Toshiyuki. 2011. Stable spectrum for pseudo- Riemannian locally symmetric spaces. Comptes Rendus Mathématique. Académie des Sciences. Paris, 349, 29–33.Google Scholar
[22] Kassel, Fanny, and Kobayashi, Toshiyuki. 2012. Discrete spectrum for non-Riemannian locally symmetric spaces. I. Construction and stability. arXiv:1209.4075.
[23] Klingler, Bruno. 1996. Complétude des variétés Lorentziennes à courbure constante. Mathematische Annalen, 306(2), 353–370.CrossRefGoogle Scholar
[24] Kobayashi, Toshiyuki. 1989. Proper action on a homogeneous space of reductive type. Mathematische Annalen, 285, 249–263.CrossRefGoogle Scholar
[25] Kobayashi, Toshiyuki. 1992a. Discontinuous groups acting on homogeneous spaces of reductive type. Pages 59–75 of: Kawazoe, T., Oshima, T., and Sano, S. (eds), Representation Theory of Lie Groups and Lie Algebras at Fuji-Kawaguchiko, 1990 August September. World Scientific.Google Scholar
[26] Kobayashi, Toshiyuki. 1992b. A necessary condition for the existence of compact Clifford-Klein forms of homogeneous spaces of reductive type. Duke Mathematical Journal, 67, 653–664.Google Scholar
[27] Kobayashi, Toshiyuki. 1996a. Criterion for proper action on homogeneous spaces of reductive groups. Journal of Lie Theory, 6(2), 147–163.Google Scholar
[28] Kobayashi, Toshiyuki. 1996b. Discontinuous groups and Clifford-Klein forms of pseudo-Riemannian homogeneous manifolds. Pages 99–165 of: Schlichtkrull, H., and Ørsted, B.(eds), Algebraic and Analytic Methods in Representation Theory. Perspectives in Mathematics, vol. 17. Academic Press.
[29] Kobayashi, Toshiyuki. 1998. Deformation of compact Clifford-Klein forms of indefinite-Riemannian homogeneous manifolds. Mathematische Annalen, 310, 395–409.CrossRefGoogle Scholar
[30] Kobayashi, Toshiyuki. 2001. Discontinuous groups for non-Riemannian homogeneous spaces. Pages 723–747 of: Engquist, B., and Schmid, W. (eds), Mathematics Unlimited – 2001 and Beyond. Springer.
[31] Kobayashi, Toshiyuki, and Yoshino, Taro. 2005. Compact Clifford-Klein forms of symmetric spaces – revisited. Pure and Applied Mathematics Quarterly, 1(3), 591–663.CrossRefGoogle Scholar
[32] Kulkarni, R. 1981. Proper actions and pseudo-Riemannian space forms. Advances in Mathematics, 40, 10–51.CrossRefGoogle Scholar
[33] Kulkarni, R., and Raymond, F. 1985. 3-dimensional Lorentz space-forms and Seifert fiber spaces. Journal of Differential Geometry, 21, 231–268.CrossRefGoogle Scholar
[34] Labourie, F., Mozes, S., and Zimmer, R.J. 1995. On Manifolds Locally Modelled on Non-Riemannian Homogeneous Spaces. Geometric and Functional Analysis, 5(6), 955–965.CrossRefGoogle Scholar
[35] Labourie, François. 1996. Quelques résultats récents sur les espaces localement homogènes compacts. Pages 267–283 of: Manifolds and Geometry (Pisa 1993). Sympos. Math., no. XXXVI. Cambridge: Cambridge University Press.Google Scholar
[36] Labourie, François. 2006. Anosov flows, surface groups and curves in projective space. Inventiones Mathematicae, 165(1), 51–114.CrossRefGoogle Scholar
[37] Labourie, François, and Zimmer, Robert J. 1995. On the Existence of Cocompact Lattices for SL(n)/SL(m). Mathematical Research Letters, 2, 75–77.CrossRefGoogle Scholar
[38] Margulis, Gregory. 1997. Existence of compact quotients of homogeneous spaces, measurably proper actions, and decay of matrix coefficients. Bulletin de la Société Mathématique de France, 125, 447–456.CrossRefGoogle Scholar
[39] Oh, Hee. 1998. Tempered subgroups and representations with minimal decay of matrix coefficients. Bulletin de la Société Mathématique de France, 126, 355–380.CrossRefGoogle Scholar
[40] Oh, Hee, and Witte, Dave. 2000. New examples of compact Clifford-Klein forms of homogeneous spaces ofSO(2, n). International Mathematics Research Notices, 235–251.Google Scholar
[41] Oh, Hee, and Witte, Dave. 2002. Compact Clifford-Klein forms of homogeneous spaces of SO(2, n). Geometriae Dedicata, 89, 25–57.CrossRefGoogle Scholar
[42] Oniščik, A. L. 1969. Decompositions of reductive Lie groups. Mathematics of the USSR-Sbornik, 9(4), 515–554.
[43] Ratner, Marina. 1991. On Ragunathan's measure conjecture. Annals of Mathematics, 134(3), 545–607.CrossRefGoogle Scholar
[44] Salein, François. 1997. Variétés anti-de Sitter de dimension 3 possédant un champ de Killing non trivial. Comptes Rendus Mathématique. Académie des Sciences. Paris, 324, 525–530.Google Scholar
[45] Salein, François. 2000. Variétés anti-de Sitter de dimension 3 exotiques. Annales de l'institut Fourier, 50(1), 257–284.CrossRefGoogle Scholar
[46] Serre, J.-P. 1971. Cohomologie des groupes discrets. Pages 337–350 of: Séminaire Bourbaki, 23`eme année (1970/1971), Exp. No. 399. Lecture Notes in Math, vol. 244. Springer.Google Scholar
[47] Shalom, Yehuda. 2000. Rigidity, unitary representations of semisimple groups, and fundamental groups of manifolds with rank one transformation group. Annals of Mathematics, 152, 113–182.CrossRefGoogle Scholar
[48] Thurston, W. P. 1980. Geometry and topology of three-manifolds. (unpublished; available from library.msri.org/books/gt3m/).
[49] Weil, André. 1964. Remarks on the cohomology of groups. Annals of Mathematics, 80, 149–157.CrossRefGoogle Scholar
[50] Wolf, Joseph A. 1962. The Clifford-Klein space forms of indefinite metric. Annals of Mathematics, 75, 77–80.CrossRefGoogle Scholar
[51] Wolf, Joseph A. 2011. Spaces of constant curvature. 6th edn. Providence, RI: AMS Chelsea Publishing.Google Scholar
[52] Zimmer, Robert J. 1984. Ergodic theory and semisimple groups. Monographs in Mathematics, vol. 81. Basel: Birkhäuser.CrossRefGoogle Scholar
[53] Zimmer, Robert J. 1994. Discrete Groups and Non-Riemannian Homogeneous Spaces. Journal of the American Mathematical Society, 7(1), 159–168.CrossRefGoogle Scholar

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