Mathematics – Geometric Topology
Scientific paper
2009-04-13
Mathematics
Geometric Topology
41 pages, 26 figures. Added a section on the PSL(2,C) A-polynomial and reorganised the results
Scientific paper
Given an ideal triangulation of a connected 3-manifold with non-empty 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 the gluing equations, from which 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 every orientable 3-manifold with torus boundary components there exists a triangulation such that we can recover any irreducible representation whose image is not a generalised dihedral group from the associated extended deformation variety. As an application, we show that this extended deformation variety detects all factors of the PSL(2,C) A-polynomial associated to the components consisting of the representations it recovers.
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