Geometry & Topology

The Maskit embedding of the twice punctured torus

Caroline Series

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Abstract

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

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

Dates
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
https://projecteuclid.org/euclid.gt/1513883511

Digital Object Identifier
doi:10.2140/gt.2010.14.1941

Mathematical Reviews number (MathSciNet)
MR2680208

Zentralblatt MATH identifier
1207.30070

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

Keywords
Kleinian group Maskit embedding bending lamination pleating ray representation variety

Citation

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. https://projecteuclid.org/euclid.gt/1513883511


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