## Geometry & Topology

### The Maskit embedding of the twice punctured torus

Caroline Series

#### Abstract

The Maskit embedding $ℳ$ 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 $ℳ$ 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 $ℳ$ 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 $ℳ$. Our main result is a formula for the asymptotic direction of $P$ in $ℳ$ as the bending measure tends to zero, in terms of natural parameters for the complex $2$–dimensional representation space $ℛ$ 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 $ℳ$ in $ℛ$.

#### Article information

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

Dates
Revised: 30 June 2010
Accepted: 3 June 2010
First available in Project Euclid: 21 December 2017

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

#### 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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