The Maskit embedding of the twice punctured torus

Abstract

The Maskit embedding $\mathcal{ℳ}$ 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 $\mathcal{T}\left(\Sigma \right)$.

We investigate $\mathcal{ℳ}$ 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 ${\mathcal{P}}_{\left[\mu \right]}$ consists of those groups in $\mathcal{ℳ}$ 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 $\mathcal{P}$ is a real $1$–submanifold of $\mathcal{ℳ}$. Our main result is a formula for the asymptotic direction of $\mathcal{P}$ in $\mathcal{ℳ}$ as the bending measure tends to zero, in terms of natural parameters for the complex $2$–dimensional representation space $\mathcal{ℛ}$ 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 $\mathcal{ℳ}$ in $\mathcal{ℛ}$.