Earle Slices Associated with Involutions for Once Punctured Torus

Abstract

In this paper, we will study Earle slices of quasi-fuchsian space for once punctured torus associated with involutions of its fundamental group induced by orientation reversing diffeomorphism of this surface. First we classify Earle slices into two types: rhombic Earle slices and rectangular Earle slices. The main purpose of this paper is to study the configuration of Earle slices. Especially, we obtain a necessary and sufficient condition for two Earle slices to intersect each other. We also show that the union of all Earle slices is connected. In the end, we describe Earle slices by using trace coordinates of quasi-fuchsian space.