# zbMATH — the first resource for mathematics

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 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
Full Text: