Volume 14, issue 4 (2010)

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The Maskit embedding of the twice punctured torus

Caroline Series

Geometry & Topology 14 (2010) 1941–1991
Abstract

The Maskit embedding $\mathsc{ℳ}$ of a surface $\Sigma$ is the space of geometrically finite groups on the boundary of quasifuchsian space for which the “top” end is homeomorphic to $\Sigma$, while the “bottom” end consists of triply punctured spheres, the remains of $\Sigma$ 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 $\mathsc{T}\left(\Sigma \right)$.

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

Keywords
Kleinian group, Maskit embedding, bending lamination, pleating ray, representation variety
Mathematical Subject Classification 2000
Primary: 30F40
Secondary: 30F60, 57M50
Publication
Received: 6 January 2009
Revised: 30 June 2010
Accepted: 3 June 2010
Published: 29 August 2010
Proposed: Jean-Pierre Otal
Seconded: Colin Rourke, Joan Birman
Authors
 Caroline Series Mathematics Institute University of Warwick Coventry CV4 7AL UK http://www.warwick.ac.uk/~masbb