Abstract
Let $\Gamma_0\subset \mathrm{SO}(n,1)$ $(n\ge 2)$ be a cocompact lattice and $\rho\colon \Gamma_0\to \Gamma$ be an injective representation into a convex-cocompat discrete isometric subgroup of a noncompact rank-$1$ symmetric space. The Hausdorff dimension $\delta(\Gamma)$ of the limit set of $\Gamma = \rho(\Gamma_0)$ satisfied $\delta(\Gamma) \ge \delta(\Gamma_0) = n-1$. We prove that equality holds if and only if $\rho$ is a Fuchsian representation; i.e., $\Gamma$ preserves a totoally geodesic copy of $H_{\mathbb{R}}^n$ in $H_{\mathbb{R}}^m$. This generalizes the results of [2] and settes a question raised by Tukia ([43], 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 complext hyperbolic geometry. That is, for a cocompact lattice $\Gamma_0 \subset \mathrm{SU}(n,1)$ $(n\ge 2)$ and an injective representation $\rho\colon \Gamma_0\to \mathrm{SU}(m,1)$ $(n\le m\le 2n-1)$ 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 $H_{\mathbb{C}}^n$ in $H_{\mathbb{C}}^m$. This can be viewed as a global generalization of Goldman and Millson’s local rigidity theorem (see [20]; another global generalization was obtained by K. Corlette [5]). Various other related rigidity results are also obtained.
