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2012 A generalisation of the deformation variety
Henry Segerman
Algebr. Geom. Topol. 12(4): 2179-2244 (2012). DOI: 10.2140/agt.2012.12.2179


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.


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Henry Segerman. "A generalisation of the deformation variety." Algebr. Geom. Topol. 12 (4) 2179 - 2244, 2012.


Received: 26 January 2011; Revised: 16 June 2012; Accepted: 23 July 2012; Published: 2012
First available in Project Euclid: 19 December 2017

zbMATH: 1260.57030
MathSciNet: MR3020204
Digital Object Identifier: 10.2140/agt.2012.12.2179

Primary: 57M50

Keywords: 3–manifold , A–polynomial , character variety , gluing equations , Hyperbolic , ideal triangulation

Rights: Copyright © 2012 Mathematical Sciences Publishers


Vol.12 • No. 4 • 2012
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