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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

MSC:
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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