Dimension and rigidity of quasi-Fuchsian representations.

*(English)*Zbl 0843.22019Summary: 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 Étud. 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.

##### MSC:

22E46 | Semisimple Lie groups and their representations |

30F40 | Kleinian groups (aspects of compact Riemann surfaces and uniformization) |

32Q45 | Hyperbolic and Kobayashi hyperbolic manifolds |