### Computing Varieties of Representations of Hyperbolic $3$-Manifolds into $\SLfR$

Daryl Cooper, Darren Long, and Morwen Thistlethwaite
Source: Experiment. Math. Volume 15, Issue 3 (2006), 291-306.

#### Abstract

The geometric structure on a closed orientable hyperbolic 3-manifold determines a discrete faithful representation $\rho$ of its fundamental group into $\mathrm{SO^{+}(3,1)}$, unique up to conjugacy. Although Mostow rigidity prohibits us from deforming $\rho$, we can try to deform the composition of $\rho$ with inclusion of $\mathrm{SO^{+}(3,1)}$ 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 $\rho$ deforms into $\mathrm{SL(4,\mathbb R)}$, 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.

First Page:
Primary Subjects: 57M50
Secondary Subjects: 57-04
Full-text: Open access