On the character variety of the four-holed sphere

TL;DR

This paper investigates the structure of the SL(2,C) character varieties of the four-holed sphere, analyzing the mapping class group action and identifying a domain of discontinuity, especially for real characters with non-maximal Euler class.

Contribution

It characterizes the domain of discontinuity for the mapping class group action on the character variety, including cases with non-maximal Euler class for real characters.

Findings

Identified a domain of discontinuity for the group action.

Showed the domain can be non-empty for non-maximal Euler class components.

Analyzed the structure of the character variety of the four-holed sphere.

Abstract

We study the (relative) SL(2,C) character varieties of the four-holed sphere and the action of the mapping class group on it. We describe a domain of discontinuity for this action, and, in the case of real characters, show that this domain of discontinuity may be non-empty on the components where the relative euler class is non-maximal.