Local rigidity of hyperbolic 3-manifolds after Dehn surgery

DOI: doi:10.1215/S0012-7094-02-11411-2

Abstract: It is well known that some lattices in SO(n, 1) can be nontrivially deformed when included in SO(n+1, 1) (e.g., via bending on a totally geodesic hypersurface); this contrasts with the (super) rigidity of higher rank lattices. M. Kapovich recently gave the first examples of lattices in SO(3, 1) which are locally rigid in SO(4, 1) by considering closed hyperbolic 3-manifolds obtained by Dehn filling on hyperbolic two-bridge knots. We generalize this result to Dehn filling on a more general class of one-cusped finite volume hyperbolic 3-manifolds, allowing us to produce the first examples of closed hyperbolic 3-manifolds which contain embedded quasi-Fuchsian surfaces but are locally rigid in SO(4, 1).