On dynamics of the Mapping class group action on relative $\text{PSL}(2,\mathbb{R})$-Character Varieties

TL;DR

This paper investigates the action of the mapping class group on relative PSL(2,R)-character varieties of punctured surfaces, introducing simple-stability as an analogue to primitive stability and identifying domains of discontinuity.

Contribution

It defines simple-stability for representations into PSL(2,R), proves these form domains of discontinuity, and provides examples of primitive-stable representations from hyperbolic cone surfaces.

Findings

Holonomies of hyperbolic cone surfaces are simple-stable.

Holonomies of cone surfaces with one cone-point of angle less than π are primitive-stable.

Provides infinite families of primitive-stable representations.

Abstract

In this paper, we study the mapping class group action on the relative $\text{PSL}(2,\mathbb{R})$-character varieties of punctured surfaces. It is well known that Minsky's primitive-stable representations form a domain of discontinuity for the $\text{Out}(F_n)$-action on the $\text{PSL}(2,\mathbb{C})$-character variety. We define simple-stability of representations of fundamental group of a surface into $\text{PSL}(2,\mathbb{R})$ which is an analogue of the definition of primitive stability and prove that these representations form a domain of discontinuity for the $\text{MCG}$-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.