Algebraic & Geometric Topology

A generalisation of the deformation variety

Henry Segerman

Full-text: Open access

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

Article information

Source
Algebr. Geom. Topol., Volume 12, Number 4 (2012), 2179-2244.

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

Permanent link to this document
https://projecteuclid.org/euclid.agt/1513715454

Digital Object Identifier
doi:10.2140/agt.2012.12.2179

Mathematical Reviews number (MathSciNet)
MR3020204

Zentralblatt MATH identifier
1260.57030

Subjects
Primary: 57M50: Geometric structures on low-dimensional manifolds

Keywords
ideal triangulation 3–manifold hyperbolic gluing equations character variety A–polynomial

Citation

Segerman, Henry. A generalisation of the deformation variety. Algebr. Geom. Topol. 12 (2012), no. 4, 2179--2244. doi:10.2140/agt.2012.12.2179. https://projecteuclid.org/euclid.agt/1513715454


Export citation

References

  • F Bruhat, J Tits, Groupes réductifs sur un corps local, Inst. Hautes Études Sci. Publ. Math. (1972) 5–251
  • A Champanerkar, A–polynomial and Bloch invariants of hyperbolic $3$–manifolds, Ph.d. thesis, Columbia University (2003)
  • D Cooper, M Culler, H Gillet, D D Long, P B Shalen, Plane curves associated to character varieties of $3$–manifolds, Invent. Math. 118 (1994) 47–84
  • M Culler, personal communication (2012)
  • M Culler, N Dunfield, J R Weeks, SnapPy, a computer program for studying the geometry and topology of $3$–manifolds Available at \setbox0\makeatletter\@url http://snappy.computop.org {\unhbox0
  • M Culler, P B Shalen, Varieties of group representations and splittings of $3$–manifolds, Ann. of Math. 117 (1983) 109–146
  • S Garoufalidis, C Koutschan, The non-commutative A-polynomial of $(-2,3,n)$ pretzel knots (2012)
  • S Garoufalidis, T W Mattman, The $A$–polynomial of the $(-2,3,3+2n)$ pretzel knots, New York J. Math. 17 (2011) 269–279
  • F Guéritaud, On canonical triangulations of once-punctured torus bundles and two-bridge link complements, Geom. Topol. 10 (2006) 1239–1284
  • T W Mattman, The Culler–Shalen seminorms of the $(-2,3,n)$ pretzel knot, J. Knot Theory Ramifications 11 (2002) 1251–1289
  • S Matveev, Algorithmic topology and classification of $3$–manifolds, 2nd edition, Algorithms and Computation in Mathematics 9, Springer, Berlin (2007)
  • T Ohtsuki, How to construct ideal points of ${\rm SL}\sb 2({\mathbb C})$ representation spaces of knot groups, Topology Appl. 93 (1999) 131–159
  • H Segerman, Detection of incompressible surfaces in hyperbolic punctured torus bundles, Geom. Dedicata 150 (2011) 181–232
  • H Segerman, S Tillmann, Pseudo-developing maps for ideal triangulations I: essential edges and generalised hyperbolic gluing equations, from: “Topology and geometry in dimension three”, (W Li, L Bartolini, J Johnson, F Luo, R Myers, J H Rubinstein, editors), Contemp. Math. 560, Amer. Math. Soc., Providence, RI (2011) 85–102
  • J-P Serre, Trees, Springer Monographs in Mathematics, Springer, Berlin (2003)
  • P B Shalen, Representations of $3$–manifold groups, from: “Handbook of geometric topology”, (R J Daverman, R B Sher, editors), North-Holland, Amsterdam (2002) 955–1044
  • W P Thurston, The geometry and topology of three-manifolds, Princeton Univ. Math. Dept. Lecture Notes (1979) Available at \setbox0\makeatletter\@url http://msri.org/publications/books/gt3m/ {\unhbox0
  • S Tillmann, Normal surfaces in topologically finite $3$–manifolds, Enseign. Math. 54 (2008) 329–380
  • S Tillmann, Degenerations of ideal hyperbolic triangulations, Math. Z. 272 (2012) 793–823
  • J Weeks, SnapPea, a computer program for creating and studying hyperbolic 3–manifolds Available at \setbox0\makeatletter\@url http://www.geometrygames.org/SnapPea/ {\unhbox0
  • T Yoshida, On ideal points of deformation curves of hyperbolic $3$–manifolds with one cusp, Topology 30 (1991) 155–170