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Computing varieties of representations of hyperbolic 3-manifolds into SL(4, R) Daryl Cooper1

Summary: Computing varieties of representations of hyperbolic 3-manifolds into SL(4, R)
Daryl Cooper1
, Darren Long2
and Morwen Thistlethwaite
1. Introduction
Following the seminal work of M. Culler and P. Shalen [Culler and Shalen, 1983], and that of
A. Casson [Akbulut and 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 and 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
(3, 1)-representation 0 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


Source: Akhmedov, Azer - Department of Mathematics, University of California at Santa Barbara


Collections: Mathematics