On the boundary of the Earle slice for punctured torus groups

Summary

Abstract

We shall show that the Earle slice ℰ for punctured torus groups is a Jordan domain, and every pleating ray in ℰ lands at a unique boundary group whose pair of end invariants is equal to the pleating invariants of this ray. We will also study the asymptotic behavior of the boundary of ℰ.

Introduction

In [Min99], Minsky showed that any marked punctured torus group can be characterized by its pair of end invariants, where a punctured torus group is a rank two free Kleinian group whose commutator of generators is parabolic. To prove this result, called the Ending Lamination Theorem, he also proved another important result, called the Pivot Theorem, which controls thin parts of the corresponding hyperbolic manifold from the data of end invariants. As one of applications of these theorems, he showed that the Bers slice and the Maskit slice are Jordan domains.

In this paper we apply his results to the Earle slice which is a holomorphic slice of quasi-fuchsian space representing the Teichmüller space of once-punctured tori. This slice was considered originally by Earle in [Ear81], and its geometric coordinates, named pleating coordinates was studied by Series and the author in [KoS01]. By using rational pleating rays, the figure of the Earle slice ℰ realized in the complex plane ℂ was drawn by Liepa. (See Figure 1. In fact only the upper half of the Earle slice is shown, the picture being symmetrical under reflection in the real axis.)