A generalisation of the deformation variety

Abstract

Given an ideal triangulation of a connected $3$–manifold 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 $3$–dimensional 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 $3$–manifold 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 $3$–manifold 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 $PSL\left(2,ℂ\right)$ A–polynomial associated to the components consisting of the representations it recovers.