Geometry & Topology

The Maskit embedding of the twice punctured torus

Caroline Series

Full-text: Open access


The Maskit embedding of a surface Σ is the space of geometrically finite groups on the boundary of quasifuchsian space for which the “top” end is homeomorphic to Σ, while the “bottom” end consists of triply punctured spheres, the remains of Σ when a set of pants curves have been pinched. As such representations vary in the character variety, the conformal structure on the top side varies over the Teichmüller space T(Σ).

We investigate when Σ is a twice punctured torus, using the method of pleating rays. Fix a projective measure class [μ] supported on closed curves on Σ. The pleating ray P[μ] consists of those groups in for which the bending measure of the top component of the convex hull boundary of the associated 3–manifold is in [μ]. It is known that P is a real 1–submanifold of . Our main result is a formula for the asymptotic direction of P in as the bending measure tends to zero, in terms of natural parameters for the complex 2–dimensional representation space and the Dehn–Thurston coordinates of the support curves to [μ] relative to the pinched curves on the bottom side. This leads to a method of locating in .

Article information

Geom. Topol., Volume 14, Number 4 (2010), 1941-1991.

Received: 6 January 2009
Revised: 30 June 2010
Accepted: 3 June 2010
First available in Project Euclid: 21 December 2017

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 30F40: Kleinian groups [See also 20H10]
Secondary: 30F60: Teichmüller theory [See also 32G15] 57M50: Geometric structures on low-dimensional manifolds

Kleinian group Maskit embedding bending lamination pleating ray representation variety


Series, Caroline. The Maskit embedding of the twice punctured torus. Geom. Topol. 14 (2010), no. 4, 1941--1991. doi:10.2140/gt.2010.14.1941.

Export citation


  • A Austin, Visualising the Maskit embedding of the twice punctured torus, MMath project, University of Warwick (2010)
  • A Austin, C Series, Computing pleating rays for the twice punctured torus, in preparation
  • A F Beardon, The geometry of discrete groups, Graduate Texts in Math. 91, Springer, New York (1983)
  • J S Birman, C Series, Algebraic linearity for an automorphism of a surface group, J. Pure Appl. Algebra 52 (1988) 227–275
  • F Bonahon, J-P Otal, Laminations measurées de plissage des variétés hyperboliques de dimension 3, Ann. of Math. $(2)$ 160 (2004) 1013–1055
  • R D Canary, D B A Epstein, P Green, Notes on notes of Thurston, from: “Analytical and geometric aspects of hyperbolic space (Coventry/Durham, 1984)”, (D B A Epstein, editor), London Math. Soc. Lecture Note Ser. 111, Cambridge Univ. Press (1987) 3–92
  • Y Chiang, Geometric intersection numbers on a five-punctured sphere, Ann. Acad. Sci. Fenn. Math. 26 (2001) 73–124
  • Y-E Choi, K Rafi, C Series, Lines of minima and Teichmüller geodesics, Geom. Funct. Anal. 18 (2008) 698–754
  • Y-E Choi, C Series, Lengths are coordinates for convex structures, J. Differential Geom. 73 (2006) 75–117
  • D B A Epstein, A Marden, Convex hulls in hyperbolic space, a theorem of Sullivan, and measured pleated surfaces, from: “Analytical and geometric aspects of hyperbolic space (Coventry/Durham, 1984)”, (D B A Epstein, editor), London Math. Soc. Lecture Note Ser. 111, Cambridge Univ. Press (1987) 113–253
  • M Kapovich, Hyperbolic manifolds and discrete groups, Progress in Math. 183, Birkhäuser, Boston (2001)
  • L Keen, B Maskit, C Series, Geometric finiteness and uniqueness for Kleinian groups with circle packing limit sets, J. Reine Angew. Math. 436 (1993) 209–219
  • L Keen, J R Parker, C Series, Pleating coordinates for the Maskit embedding of the Teichmüller space of punctured tori, Unpublished draft manuscript (1996)
  • L Keen, J R Parker, C Series, Combinatorics of simple closed curves on the twice punctured torus, Israel J. Math. 112 (1999) 29–60
  • L Keen, C Series, Pleating coordinates for the Maskit embedding of the Teichmüller space of punctured tori, Topology 32 (1993) 719–749
  • L Keen, C Series, Pleating invariants for punctured torus groups, Topology 43 (2004) 447–491
  • I Kra, Horocyclic coordinates for Riemann surfaces and moduli spaces. I. Teichmüller and Riemann spaces of Kleinian groups, J. Amer. Math. Soc. 3 (1990) 499–578
  • S Maloni, C Series, Top terms of trace polynomials in Kra's plumbing construction, Alg. Geom. Topol. 10 (2010) 1565–1607
  • A Marden, Outer circles. An introduction to hyperbolic $3$–manifolds, Cambridge Univ. Press (2007)
  • B Maskit, Kleinian groups, Grund. der Math. Wissenschaften 287, Springer, Berlin (1988)
  • C McMullen, Cusps are dense, Ann. of Math. $(2)$ 133 (1991) 217–247
  • J Milnor, Singular points of complex hypersurfaces, Annals of Math. Studies 61, Princeton Univ. Press (1968)
  • Y N Minsky, Extremal length estimates and product regions in Teichmüller space, Duke Math. J. 83 (1996) 249–286
  • H Miyachi, Cusps in complex boundaries of one-dimensional Teichmüller space, Conform. Geom. Dyn. 7 (2003) 103–151
  • D Mumford, C Series, D Wright, Indra's pearls. The vision of Felix Klein, Cambridge Univ. Press (2002)
  • R C Penner, J L Harer, Combinatorics of train tracks, Annals of Math. Studies 125, Princeton Univ. Press (1992)
  • C Series, An extension of Wolpert's derivative formula, Pacific J. Math. 197 (2001) 223–239
  • C Series, Limits of quasi-Fuchsian groups with small bending, Duke Math. J. 128 (2005) 285–329
  • D Thurston, On geometric intersection of curves in surfaces Available at \setbox0\makeatletter\@url {\unhbox0
  • D J Wright, The shape of the boundary of Maskit's embedding of the Teichmüller space of once punctured tori, Unpublished preprint (1988)