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

Ajay Kumar Nair

TL;DR

The paper investigates the action of the mapping class group $\mathrm{MCG}(S_{g,n})$ on relative $\mathrm{PSL}(2,\mathbb{R})$-character varieties, introducing simple-stability as an analogue of primitive stability for punctured surfaces. It proves that the simple-stable representations form a domain of discontinuity for the $\mathrm{MCG}$-action and shows that holonomies of admissible hyperbolic cone surfaces are simple-stable, with single-cone-point holonomies being primitive-stable, yielding infinite families of indiscrete primitive-stable representations. The authors also develop strong simple stability for punctured spheres and establish SBQ-conditions, providing a robust framework (including for cone-angles $<\pi$) that links geometric structures to dynamical properties of the group action. These results extend Minsky’s primitive-stable framework to relative PSL$(2,\mathbb{R})$-varieties and identify new domains of discontinuity and geometric examples, deepening the understanding of how hyperbolic cone geometries encode representation-theoretic behavior.

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 $π$ are primitive-stable, thus giving examples of an infinite family of indiscrete primitive-stable representations.

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

TL;DR

The paper investigates the action of the mapping class group on relative -character varieties, introducing simple-stability as an analogue of primitive stability for punctured surfaces. It proves that the simple-stable representations form a domain of discontinuity for the -action and shows that holonomies of admissible hyperbolic cone surfaces are simple-stable, with single-cone-point holonomies being primitive-stable, yielding infinite families of indiscrete primitive-stable representations. The authors also develop strong simple stability for punctured spheres and establish SBQ-conditions, providing a robust framework (including for cone-angles ) that links geometric structures to dynamical properties of the group action. These results extend Minsky’s primitive-stable framework to relative PSL-varieties and identify new domains of discontinuity and geometric examples, deepening the understanding of how hyperbolic cone geometries encode representation-theoretic behavior.

Abstract

In this paper, we study the mapping class group action on the relative -character varieties of punctured surfaces. It is well known that Minsky's primitive-stable representations form a domain of discontinuity for the -action on the -character variety. We define simple-stability of representations of fundamental group of a surface into which is an analogue of the definition of primitive stability and prove that these representations form a domain of discontinuity for the -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 are primitive-stable, thus giving examples of an infinite family of indiscrete primitive-stable representations.

Paper Structure

This paper contains 8 sections, 27 theorems, 31 equations, 10 figures.

Table of Contents

  1. Introduction
  2. Hyperbolic Cone Surfaces
  3. $\mathrm{PSL}(2,\mathbb{R})$-Character Variety and Mapping Class group action
  4. Primitive Stability and Simple Stability
  5. Holonomies of Hyperbolic Cone Surfaces
  6. Proof of Theorem 1.2
  7. Strong Simple Stability
  8. Simple Bowditch Q-Conditions

Key Result

Theorem 1.1

$\textup{MCG}(S_{g,n})$ acts properly discontinuously on $\mathcal{SS}(S_{g,n})$. In particular, $\textup{MCG}(S_{g,n})$ acts properly discontinuously on $\mathcal{SS}(S_{g,n}) \cap \mathcal{X}(S_{g,n}, \mathcal{C})$, where $\mathcal{C}$ is a set of $n$ conjugacy classes in $\mathrm{PSL}(2,\mathbb{R

Figures (10)

  • Figure 1: A cone sphere with 3 cone-points obtained by doubling the geodesic triangle on the left.
  • Figure 2: $\pi: \widetilde{\mathcal{C}^\ast_{h,\theta}} \rightarrow \mathcal{C}^\ast_{h,\theta}$
  • Figure 3: Shortest paths joining $\widetilde{x}, \widetilde{y}$
  • Figure 4: The geodesic $\gamma$ in $\mathcal{C}_{h,\theta}$ and $\mathcal{S}_{h,\theta}$.
  • Figure 5: The cone neighbourhood $U_d$.
  • ...and 5 more figures

Theorems & Definitions (75)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Definition 2.1
  • Example 2.2
  • Definition 2.3
  • Proposition 2.4
  • proof
  • Proposition 2.5
  • ...and 65 more