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Dimension and rigidity of quasi-Fuchsian representations. (English) Zbl 0843.22019
Summary: Let \(\Gamma_0 \subset \text{SO} (n,1)\) be a cocompact lattice and \(\rho : \Gamma_0 \to \Gamma\) an injective representation into a convex-cocompact discrete isometric subgroup of a noncompact rank-1 symmetric space. The Hausdorff dimension \(\delta(\Gamma)\) of the limit set of \(\Gamma = \rho(\Gamma_0)\) satisfies \(\delta(\Gamma) \geq \delta(\Gamma_0) = n - 1\). We prove that equality holds if and only if \(\rho\) is a Fuchsian representation; i.e., \(\Gamma\) preserves a totally geodesic copy of \(H^n_\mathbb{R}\) in \(H^m_\mathbb{R}\). This generalizes the result of R. Bowen [Publ. Math., Inst. Hautes √Čtud. Sci. 50, 11-25 (1979; Zbl 0439.30032)] and settles a question raised by P. Tukia [Invent. Math. 97, No. 2, 405-431 (1989; Zbl 0674.30038); p. 428]. Actually we prove a more general result in the context of variable negative curvature. Strikingly, there are no quasi-Fuchsian representations at least for the lower codimensional case in complex hyperbolic geometry. That is, for a cocompact lattice \(\Gamma_0 \subset \text{SU}(n,1)\) \((n \geq 2)\) and an injective representation \(\rho : \Gamma_0 \to \text{SU}(m,1)\) \((n \leq m \leq 2n - 1)\) with \(\Gamma = \rho(\Gamma_0)\) convex-cocompact, we prove that one always has \(\delta(\Gamma) = \delta(\Gamma_0)\) and moreover, \(\Gamma\) must stabilize a totally geodesic copy of \(H^n_\mathbb{C}\) in \(H^m_\mathbb{C}\). This can be viewed as a global generalization of W. M. Goldman and J. J. Millson’s local rigidity theorem (see [Invent. Math. 88, 495-520 (1987; Zbl 0627.22012)]; another global generalization was obtained by K. Corlette [J. Differ. Geom. 28, No. 3, 361-382 (1988; Zbl 0676.58007)]). Various other related rigidity results are also obtained.

22E46 Semisimple Lie groups and their representations
30F40 Kleinian groups (aspects of compact Riemann surfaces and uniformization)
32Q45 Hyperbolic and Kobayashi hyperbolic manifolds
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