Dimension and rigidity of quasi-Fuchsian representations

• 成果类型: Article
• 署名作者: Yue, CB
• 刊物名称: ANNALS OF MATHEMATICS
• ISSN/ISSBN: 0003-486X
• DOI: 10.2307/2118646
• 发表日期: 1996
• 页码: 331-355
• 关键词: hyperbolic space; symmetric-spaces; kleinian-groups; MANIFOLDS; geometry; superrigidity; DYNAMICS; THEOREM; entropy

摘要

Let Gamma(0) subset of SO(N, 1) (n greater than or equal to 2 be a cocompact lattice and rho: Gamma(0) --> Gamma be 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) greater than or equal to 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-R(n) in H-R(m). This generalizes the result of [2] and settles 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 complex hyperbolic geometry. That is, for a cocompact lattice Gamma(0) subset of SU(n, 1) (n greater than or equal to 2) and an injective representation rho: Gamma(0) --> SU(m, 1) (n less than or equal to m less than or equal to 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-C(n) in H-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.