The Maskit embedding of the twice punctured torus

TL;DR

This paper studies the Maskit embedding of a twice punctured torus, analyzing the asymptotic behavior of pleating rays in the space of geometrically finite groups using Dehn-Thurston coordinates.

Contribution

It provides a formula for the asymptotic direction of pleating rays in the Maskit embedding, linking geometric limits to representation parameters and Dehn-Thurston coordinates.

Findings

Derived an explicit asymptotic formula for pleating rays.

Connected pleating ray behavior to Dehn-Thurston coordinates.

Developed a method to locate the Maskit embedding in representation space.

Abstract

The Maskit embedding 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 two triply punctured spheres, the remains of \Sigma when two fixed disjoint curves have been pinched. As such representations vary in the character variety, the conformal structure on the top side varies over the Teichm\"uller space T(\Sigma). We investigate M when \Sigma is a twice punctured torus, using the method of pleating rays. Fix a projective measure class [\mu] supported on closed curves on \Sigma. The pleating ray P_[\mu] 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 [\mu]. 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 2-complex dimensional representation space R and the Dehn-Thurston coordinates of the support curves to [\mu] relative to the pinched curves on the bottom side. This leads to a method of locating M in R.