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Local rigidity of discrete groups acting on complex hyperbolic space. (English) Zbl 0627.22012
According to the superrigidity theorem, there are no non-trivial deformations of finite-dimensional representations of irreducible lattices in semisimple Lie groups of real rank strictly larger than 1. On the other hand, Johnson, Milson and Kourouniotis constructed such deformations for certain representations of lattices in SO(n,1). More exactly, it was proved that the deformation space is non-trivial for the representation of a lattice \(\Gamma\subset SO(n,1)\) obtained by restricting an inclusion \(SO(n,1)\to SO(n+1,1)\). In the reviewed paper, the complex analogue of the above example is studied.
The main result is the following Theorem 1. Let \(D^{n+1}\) be the unit ball in \({\mathbb{C}}^{n+1}\) equipped with the metric of constant holomorphic sectional curvature -1 (complex hyperbolic space). Let \(\Gamma\) be a torsion free group acting isometrically and properly discontinuously on \(D^{n+1}\) in such a way as to stabilize a totally geodesic n-ball \(D^ n\). Assume the quotient \(D^ n/\Gamma\) is compact. Then all nearby isometric actions of \(\Gamma\) on \(D^{n+1}\) also stabilize a totally geodesic n-ball.
Taking into account the local rigidity theorem, we see from Theorem 1 that the situation for lattices in SU(n,1) is opposite to the situation for lattices in SO(n,1).
Reviewer: G.A.Margulis

22E46 Semisimple Lie groups and their representations
22E40 Discrete subgroups of Lie groups
32Q45 Hyperbolic and Kobayashi hyperbolic manifolds
57S25 Groups acting on specific manifolds
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[1] Borel, A., Wallach, N.: Continuous cohomology, discrete subgroups, and representations of reductive groups. Ann. Math. Stud.94. Princeton University Press 1980 · Zbl 0443.22010
[2] Bourbaki, N.: Element de mathematiques. Groupes et algebres de Lie. Chapitres 2 et 3. Paris: Hermann 1972 · Zbl 0244.22007
[3] Faran, J.: Maps from the two-ball to the three-ball. Invent. Math.80, 441-475 (1982) · Zbl 0519.32016
[4] Goldman, W.M.: Characteristic classes and representations of discrete subgroups of Lie groups. Bull. Am. Math. Soc.6, 91-94 (1982) · Zbl 0493.57011
[5] Goldman, W.M.: Representations of fundamental groups of surfaces. In: Geometry and Topology, Proceedings, University of Maryland. Lect. Notes Math., vol. 1167, pp. 95-117. Berlin-Heidelberg-New York: Springer 1987
[6] Johnson, D., Millson, J.J.: Deformation spaces associated to compact hyperbolic manifolds. In: Discrete groups in geometry and analysis. Proceedings of a conference held at Yale University in honor of G.D. Mostow, Progress in Math. Boston: Birkhäuser 1987 · Zbl 0664.53023
[7] Kobayashi, S., Nomizu, K.: Foundations of differential geometry. Interscience tracts in pure and applied mathematics, Number 15, Vol. II
[8] Kourouniotis, C.: Deformations of hyperbolic structures on manifolds of several dimensions, Math. Proc. Camb. Phil. Soc.98, 247-261 (1985) · Zbl 0577.53041
[9] Matsushima, Y., Murakami, S.: On vector bundle valued harmonic forms and automorphic forms on symmetric spaces. Ann. Math.78 (2), 365-416 (1963) · Zbl 0125.10702
[10] Murakami, S.: Cohomology groups of vector-valued forms on symmetric spaces. Lecture notes, University of Chicago, 1966 · Zbl 0208.24402
[11] Milnor, J.: Singular points of complex hypersurfaces. Ann. Math. Stud., vol. 61. Princeton University Press 1968 · Zbl 0184.48405
[12] Raghunathan, M.S.: Discrete subgroups of Lie groups. Ergebnisse der Mathematik und ihrer Grenzgebiete, vol. 68. Berlin-Heidelberg-New York: Springer 1972 · Zbl 0254.22005
[13] Thurston, W.P.: The geometry and topology of three-manifolds. Princeton University Lecture Notes, 1978 · Zbl 0399.73039
[14] Tits, J.: Free subgroups in linear groups. J. Algebra20, 250-270 (1972) · Zbl 0236.20032
[15] Webster, S.M.: On mapping ann-ball into an (n+1)-ball in complex space. Pacific J. Math.81, 267-272 (1979) · Zbl 0379.32018
[16] Weil, A.: On discrete subgroups of Lie groups. Ann. Math.72 (2), 369-384 (1960) · Zbl 0131.26602
[17] Zimmer, R.: Ergodic theory and semisimple groups. Monographs in Mathematics. Boston: Birkhäuser, 1985 · Zbl 0598.22007
[18] Artin, M.: On the solutions of analytic equations. Invent. Math.5, 277-291 (1968) · Zbl 0172.05301
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