Mapping class group dynamics, relative character varieties and hyperbolic cone surfaces

Abstract

This thesis concerns the punctured surface group representations into $\pslr$ and $\pslc$. We study the action of mapping class group on the relative character varieties of punctured surface groups into $\pslr$ and $\pslc$. Minsky defined primitive-stable representations to find a domain of discontinuity for $\text{Out}(F_n)$ action on the $\pslc$-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 mapping class group action on $\pslc$-relative character varieties. The discrete, faithful and geometrically finite representations turn out 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 punctured surface group representations into $\pslc$. We prove that given a non-elementary representation $\rho: \pi_1(S_{g,n}) \rightarrow \pslc$ taking peripheral simple-closed curves 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 the Gallo-Kapovich-Marden paper, works with slight modifications.