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Non-ergodicity on the SU(2)-character varieties

Fayssal Saadi

TL;DR

The paper investigates when subgroups $\Gamma$ of the mapping class group, generated by Dehn twists along filling curves, act ergodically on ${\sf SU}(2)$-representation and character varieties. It connects square-tiled/origami geometry with Goldman’s flow to construct explicit $\Gamma$-invariant rational functions in genus-2 orientable and genus-4 non-orientable cases, demonstrating non-ergodicity in these settings. The approach blends dynamics on representation varieties with algebraic geometry via foliations on intersections of quadrics and yields concrete invariants in targeted examples. These results provide explicit obstructions to ergodicity, illustrating how pseudo-Anosov elements and flat-surface structures influence the dynamics of group actions on moduli spaces.

Abstract

We describe the dynamics of a group $Γ$ generated by Dehn twists along two filling multi-curves or a family of filling curves on the SU(2)-representation variety of closed surfaces. Consequently, we provide explicit $Γ$-invariant rational functions on the representation variety of the genus two closed surface $S_2$ for some pair of multi-curves. We establish a similar result for the SU(2)-character variety of genus four non-orientable surfaces $N_4$ for some family of filling curves.

Non-ergodicity on the SU(2)-character varieties

TL;DR

The paper investigates when subgroups of the mapping class group, generated by Dehn twists along filling curves, act ergodically on -representation and character varieties. It connects square-tiled/origami geometry with Goldman’s flow to construct explicit -invariant rational functions in genus-2 orientable and genus-4 non-orientable cases, demonstrating non-ergodicity in these settings. The approach blends dynamics on representation varieties with algebraic geometry via foliations on intersections of quadrics and yields concrete invariants in targeted examples. These results provide explicit obstructions to ergodicity, illustrating how pseudo-Anosov elements and flat-surface structures influence the dynamics of group actions on moduli spaces.

Abstract

We describe the dynamics of a group generated by Dehn twists along two filling multi-curves or a family of filling curves on the SU(2)-representation variety of closed surfaces. Consequently, we provide explicit -invariant rational functions on the representation variety of the genus two closed surface for some pair of multi-curves. We establish a similar result for the SU(2)-character variety of genus four non-orientable surfaces for some family of filling curves.
Paper Structure (7 sections, 16 theorems, 24 equations, 6 figures)

This paper contains 7 sections, 16 theorems, 24 equations, 6 figures.

Table of Contents

  1. Introduction
  2. multi-twists and square-tiled surfaces
  3. ${\sf{SU}}(2)$-character varieties and Goldman's flow
  4. Foliations on the intersection of quadrics
  5. Invariant functions
  6. Examples on the representation variety of $S_2$
  7. Examples on the character variety of $N_4$

Key Result

Theorem 1.1

On the closed non-orientable surface of genus four $N_4$, there exists a family of filling curves whose associated Dehn twists generate a group $\Gamma$ acting non-ergodically on the character variety $X(\pi_1(N_4),{\sf{SU}}(2))$.

Figures (6)

  • Figure 1: Two filling curves in $S_2$
  • Figure 2: Square $S_i$ in the Square-tiled surface
  • Figure 3: A rectangle in the square-tiled surface
  • Figure 4: The surface $S'$
  • Figure 5: The genus two surface $S$
  • ...and 1 more figures

Theorems & Definitions (27)

  • Theorem 1.1
  • Corollary 1.1
  • Theorem 1.2
  • Definition 2.1
  • Proposition 1
  • Proposition 2
  • Proposition 3
  • Proposition 4
  • Proposition 5
  • proof
  • ...and 17 more