A generalisation of the deformation variety
Abstract
Given an ideal triangulation of a connected 3manifold with nonempty boundary consisting of a disjoint union of tori, a point of the deformation variety is an assignment of complex numbers to the dihedral angles of the tetrahedra subject to Thurston's gluing equations. From this, one can recover a representation of the fundamental group of the manifold into the isometries of 3dimensional hyperbolic space. However, the deformation variety depends crucially on the triangulation: there may be entire components of the representation variety which can be obtained from the deformation variety with one triangulation but not another. We introduce a generalisation of the deformation variety, which again consists of assignments of complex variables to certain dihedral angles subject to polynomial equations, but together with some extra combinatorial data concerning degenerate tetrahedra. This "extended deformation variety" deals with many situations that the deformation variety cannot. In particular we show that for any ideal triangulation of a small orientable 3manifold with a single torus boundary component, we can recover all of the irreducible nondihedral representations from the associated extended deformation variety. More generally, we give an algorithm to produce a triangulation of a given orientable 3manifold with torus boundary components for which the same result holds. As an application, we show that this extended deformation variety detects all factors of the PSL(2,C) Apolynomial associated to the components consisting of the representations it recovers.
 Publication:

arXiv eprints
 Pub Date:
 April 2009
 DOI:
 10.48550/arXiv.0904.1893
 arXiv:
 arXiv:0904.1893
 Bibcode:
 2009arXiv0904.1893S
 Keywords:

 Mathematics  Geometric Topology;
 57M50
 EPrint:
 47 pages, 26 figures. Rewrote introduction and added motivation section based on referee's comments. Rewrote the section on retriangulation, and added new result on small manifolds with a single cusp