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.
Shalen, Varieties of group representations and splittings of 3 -manifolds , Ann
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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.