Computing Varieties of Representations of Hyperbolic 3-Manifolds into SL(4, ℝ)

Following the seminal work of M. Culler and P. Shalen [Culler & Shalen, 1983], and that of A. Casson [Akbulut & McCarthy, 1990], the theory of representation and character varieties of 3-manifolds has come to be recognized as a powerful tool, and has duly assumed an important place in low-dimensional topology. Among the many papers that have appeared in this context, we mention [Culler et al., 1987, Cooper et al., 1994, Boyer & Zhang, 1998]. Most of the work carried out to date is concerned with representations into Lie groups of 2 × 2 matrices, owing mainly to connections with actions on trees and the isometry groups of hyperbolic space in dimensions 2 and 3, but also owing to the extreme difficulty of computations beyond the realm of such matrices. This paper was originally motivated by the following question: under what circumstances can one take the hyperbolic structure on a closed hyperbolic 3-manifold and deform it to a real projective structure? In the language of representations, this amounts to beginning with an SO(3, 1)representation φ0 of the fundamental group of the manifold, given by the hyperbolic structure, and then endeavouring to compute the component of the SL(4, R)-representation variety containing φ0. Using a computer, it is relatively easy to see that for many closed hyperbolic 3-manifolds there are linear obstructions to deforming, but when these obstructions vanish it is of considerable interest to see whether genuine deformations exist. The