Dimension and Rigidity of Quasi-Fuchsian Representations

Abstract

Let FO C SO(n, 1) (n > 2) be a cocompact lattice and p: Fo - F be an injective representation into a convex-cocompact discrete isometric subgroup of a noncompact rank-1 symmetric space. The Hausdorff dimension 6(F) of the limit set of F = p(Fo) satisfies 6(F) > 6(Fo) = n - 1. We prove that equality holds if and only if p is a Fuchsian representation; i.e., F preserves a totally geodesic copy of HRI in HR. 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 FO C SU(n, 1) (n > 2) and an injective representation p: Fo -* SU(m, 1) (n < m < 2n - 1) with F = p(Fo) convex-cocompact, we prove that one always has 6(F) = 6(Fo) and moreover, F must stabilize a totally geodesic copy of H?: in H?:. 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.