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A generalisation of the deformation variety

Henry Segerman

Algebraic & Geometric Topology 12 (2012) 2179–2244

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(2, ) A–polynomial associated to the components consisting of the representations it recovers.

ideal triangulation, 3–manifold, hyperbolic, gluing equations, character variety, A–polynomial
Mathematical Subject Classification 2000
Primary: 57M50
Received: 26 January 2011
Revised: 16 June 2012
Accepted: 23 July 2012
Published: 26 December 2012
Henry Segerman
Department of Mathematics and Statistics
The University of Melbourne
VIC 3010