On the character variety of the four-holed sphere
Sara Maloni, Frederic Palesi, Ser Peow Tan

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.
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