On the character variety of the three-holed projective plane

TL;DR

This paper investigates the structure of the SL(2,C) character variety of the three-holed projective plane, analyzing the mapping class group action and identifying a domain of discontinuity that extends previous known sets.

Contribution

It introduces a new domain of discontinuity for the mapping class group action on the character variety, expanding beyond primitive stable and convex-cocompact representations.

Findings

Identified a domain of discontinuity larger than previously known sets.

Connected the character variety of the three-holed projective plane with that of the four-holed sphere.

Provided insights into the dynamics of the mapping class group action.

Abstract

We study the (relative) SL(2,C) character varieties of the three-holed projective plane and the action of the mapping class group on them. We describe a domain of discontinuity for this action, which strictly contains the set of primitive stable representations defined by Minsky, and also the set of convex-cocompact characters. We consider the relationship with the previous work of the authors and S. P. Tan on the character variety of the four-holed sphere.