Geometry & Topology
- Geom. Topol.
- Volume 14, Number 4 (2010), 1941-1991.
The Maskit embedding of the twice punctured torus
The Maskit embedding of a surface is the space of geometrically finite groups on the boundary of quasifuchsian space for which the “top” end is homeomorphic to , while the “bottom” end consists of triply punctured spheres, the remains of 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 .
We investigate when is a twice punctured torus, using the method of pleating rays. Fix a projective measure class supported on closed curves on . The pleating ray consists of those groups in for which the bending measure of the top component of the convex hull boundary of the associated –manifold is in . It is known that is a real –submanifold of . Our main result is a formula for the asymptotic direction of in as the bending measure tends to zero, in terms of natural parameters for the complex –dimensional representation space and the Dehn–Thurston coordinates of the support curves to relative to the pinched curves on the bottom side. This leads to a method of locating in .
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
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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