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

Abstract

The geometric structure on a closed orientable hyperbolic 3- manifold determines a discrete faithful representation ρ of its fundamental group into SO⁺(3, 1), unique up to conjugacy. Although Mostow rigidity prohibits us from deforming ρ, we can try to deform the composition of ρ with inclusion of SO⁺(3, 1) into a larger group. In this sense, we have found by exact computation a small number of closed manifolds in the Hodgson- Weeks census for which ρ deforms into SL(4,ℝ), thus showing that the hyperbolic structure can be deformed in these cases to a real projective structure. In this paper we describe the method for computing these deformations, particular attention being given to the manifold Vol3.

Keywords:

• Primary 57M50
• Secondary 57-04
• Hyperbolic 3-manifolds
• deformation of geometric structure
• algorithms