Simple-stable representations of punctured surface groups into PSL(2,R) form a domain of discontinuity for the mapping class group action, and admissible hyperbolic cone surface holonomies are simple-stable, with one-cone-point cases primitive-stable.
Computing a Dirichlet domain for a hyperbolic surface
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abstract
The goal of this paper is to exhibit and analyze an algorithm that takes a given closed orientable hyperbolic surface and outputs an explicit Dirichlet domain. The input is a fundamental polygon with side pairings. While grounded in topological considerations, the algorithm makes key use of the geometry of the surface. We introduce data structures that reflect this interplay between geometry and topology and show that the algorithm finishes in polynomial time, in terms of the initial perimeter and the genus of the surface.
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On dynamics of the Mapping class group action on relative $\text{PSL}(2,\mathbb{R})$-Character Varieties
Simple-stable representations of punctured surface groups into PSL(2,R) form a domain of discontinuity for the mapping class group action, and admissible hyperbolic cone surface holonomies are simple-stable, with one-cone-point cases primitive-stable.