On dynamics of the mapping class group action on relative $\operatorname{PSL}(2,\mathbb{R})$-character varieties

In this paper, we study the mapping class group action on the relative \operatorname{PSL}(2,\mathbb{R}) -character varieties of punctured surfaces. It is well known that Minsky’s primitive-stable representations form a domain of discontinuity for the \operatorname{Out}(F_{n}) -action on the \operatorname{PSL}(2,\mathbb{C}) -character variety. We define simple stability of representations of fundamental group of a surface into \operatorname{PSL}(2,\mathbb{R}) which is an analogue of the definition of primitive stability and prove that these representations form a domain of discontinuity for the \operatorname{MCG} -action. Our first main result shows that holonomies of hyperbolic cone surfaces are simple-stable. We also prove that holonomies of hyperbolic cone surfaces with exactly one cone-point of cone-angle less than \pi are primitive-stable, thus giving examples of an infinite family of indiscrete primitive-stable representations.