This thesis explores the dynamics of the mapping class group action on relative character varieties. We focus on the representations from the fundamental group of punctured surfaces into $\mathrm{PSL}(2,\mathbb{R})$ and $\mathrm{PSL}(2,\mathbb{C})$.

The first part of the thesis focuses on representations into $\mathrm{PSL}(2,\mathbb{R})$. We look at the relative character varieties of the fundamental group of punctured surfaces into $\mathrm{PSL}(2,\mathbb{R})$ and the action of the mapping class group on them. Minsky defined primitive-stable representations to find a domain of discontinuity for the $\mathrm{Out}(F_n)$-action on the $\mathrm{PSL}(2,\mathbb{C})$-character variety. Following this idea, we define simple-stability, which is a natural analogue of primitive-stability in the setting of mapping class group action.

We prove that simple-stable representations form a domain of discontinuity for the mapping class group action on relative character varieties. The discrete, faithful and geometrically finite representations turn out to be simple-stable. Our first main result shows that holonomies of admissible hyperbolic cone surfaces are simple-stable, thus giving indiscrete examples of simple-stable representations. 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.

The second part of the thesis concerns representations of the fundamental group of punctured surfaces into $\mathrm{PSL}(2,\mathbb{C})$. We prove that given a non-elementary representation $\rho: \pi_1(S_{g,n}) \to \mathrm{PSL}(2,\mathbb{R})$ taking peripheral simple closed curve elements to loxodromic elements, we can find a pair of pants decomposition of $S_{g,n}$ such that restriction of the representation to each of these pants is Schottky. The proof shows that the techniques used to prove this in the case of closed surfaces, as in Gallo-Kapovich-Marden, work with slight modifications.

In this talk, we will be mostly focusing on the first part of the thesis.