Experimental Mathematics

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

Daryl Cooper, Darren Long, and Morwen Thistlethwaite

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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.

Article information

Source
Experiment. Math. Volume 15, Issue 3 (2006), 291-306.

Dates
First available in Project Euclid: 5 April 2007

Permanent link to this document
http://projecteuclid.org/euclid.em/1175789760

Mathematical Reviews number (MathSciNet)
MR2264468

Zentralblatt MATH identifier
1117.57016

Subjects
Primary: 57M50: Geometric structures on low-dimensional manifolds
Secondary: 57-04: Explicit machine computation and programs (not the theory of computation or programming)

Keywords
Hyperbolic 3-manifolds deformation of geometric structure algorithms

Citation

Cooper, Daryl; Long, Darren; Thistlethwaite, Morwen. Computing Varieties of Representations of Hyperbolic $3$-Manifolds into $\SLfR$. Experiment. Math. 15 (2006), no. 3, 291--306. http://projecteuclid.org/euclid.em/1175789760.


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