Infinitesimal deformations of some SO(3,1) lattices

Abstract

Let Γ be a torsion-free lattice in SO 0 (3, 1), and let M = Γ\H 3 be the corresponding hyperbolic 3-manifold. It is wellknown that in the presence of a closed, embedded, totallygeodesic surface in M , the canonical flat conformal structure on M can be deformed via the bending construction. Equivalently, the lattice Γ admits non-trivial deformations into SO 0 (4, 1). We present a new construction of infinitesimal deformations for the hyperbolic Fibonacci manifolds, the smallest of which is non-Haken and contains no immersed totally geodesic surface.

Rigidity of representations in SO(4,1) for Dehn fillings on 2-bridge knots

“…We are interested in representations up to conjugacy, but the space of conjugacy classes does not seem to have a structure easy to work with. For results about this set of conjugacy classes we refer to [Johnson and Millson 1987;Kapovich 1994;Morgan 1986;Scannell 2000]. When = π 1 (M), it is customary to write R(M, SO 0 (n, 1)) = R π 1 (M), SO 0 (n, 1) .…”

Local rigidity of hyperbolic 3-manifolds after Dehn surgery

“…This paper continues our study of the local deformation theory of rank-one lattices, which began with [28]. We are particularly interested in the local deformation space of representations of an SO(3, 1) lattice when viewed as a "Fuchsian" subgroup of SO (4,1).…”

“…In [28] we gave examples of infinitesimal deformations for infinitely many two-generator, closed hyperbolic 3-manifolds with zero first Betti number. One of these examples is non-Haken, and its fundamental group contains no nonelementary Fuchsian subgroups, providing an infinitesimal counterexample to one half of Kapovich's conjecture.…”

A note on stamping

“…Very little else is known about this question in general, though there are a few other interesting examples in the literature; some rigid [13], and some admitting deformations [5,6,12,14] (occasionally only to first order).…”