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Pseudo-Anosov action on the $SU(2)$-character variety of $S_2$

Fayssal Saadi

TL;DR

The work addresses the action of mapping class subgroups on the ${\mathrm{SU}}(2)$-character variety by constructing a genus-2 subgroup $\\Gamma$ with infinitely many pseudo-Anosov elements that preserves a nontrivial rational function on the ${\mathrm{SU}}(2)$-character variety of $S_{2}$. It leverages a Birman–Hilden representation, lifts to the homeomorphism group via square-tiled surfaces, and a refined intersection of subgroups to produce a $\\Gamma$ with abundant pseudo-Anosov elements. The invariant function arises from two independent directions on representation spaces, whose angle remains constant under $\\Gamma$ while the function is shown to be non-constant, thus delivering a concrete answer to Goldman’s question about invariant rational functions on the character variety. Overall, the paper integrates square-tiled surface techniques, chain relations, and Fathi’s pseudo-Anosov results to produce explicit invariant structures for a nontrivial subgroup of $\\mathrm{Mod}(S_{2})$.

Abstract

In this article, we continue the study of the action of subgroups of the mapping class group on the $SU(2)$-character variety. We prove the existence of a mapping class subgroup on the surface of genus $2$, containing infinitely many pseudo-Anosov elements, which admit an invariant rational function on $SU(2)$-character variety of $S_2$.

Pseudo-Anosov action on the $SU(2)$-character variety of $S_2$

TL;DR

The work addresses the action of mapping class subgroups on the -character variety by constructing a genus-2 subgroup with infinitely many pseudo-Anosov elements that preserves a nontrivial rational function on the -character variety of . It leverages a Birman–Hilden representation, lifts to the homeomorphism group via square-tiled surfaces, and a refined intersection of subgroups to produce a with abundant pseudo-Anosov elements. The invariant function arises from two independent directions on representation spaces, whose angle remains constant under while the function is shown to be non-constant, thus delivering a concrete answer to Goldman’s question about invariant rational functions on the character variety. Overall, the paper integrates square-tiled surface techniques, chain relations, and Fathi’s pseudo-Anosov results to produce explicit invariant structures for a nontrivial subgroup of .

Abstract

In this article, we continue the study of the action of subgroups of the mapping class group on the -character variety. We prove the existence of a mapping class subgroup on the surface of genus , containing infinitely many pseudo-Anosov elements, which admit an invariant rational function on -character variety of .
Paper Structure (4 sections, 11 theorems, 28 equations, 4 figures)

This paper contains 4 sections, 11 theorems, 28 equations, 4 figures.

Table of Contents

  1. introduction
  2. Presentation of the mapping class group of the closed surface $S_2$
  3. Intersection of subgroups
  4. Invariant functions

Key Result

Theorem 1.1

The group generated by the Dehn-twists along the pair of multi-curves associated to the square-tiled surface $S$ (Figure 2 curves) admits an invariant rational function on the representation variety ${\sf{Hom}}(\pi_1(S),{\sf{SU}}(2))$.

Figures (4)

  • Figure 1: Square-tiled surface $S$
  • Figure 2: Presentation of ${\sf{Mod}}(S_2)$
  • Figure 3: Associated square-tiled surface
  • Figure 4: System of curves

Theorems & Definitions (17)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 2.1: Birman-Hilden
  • Proposition 1: $k$-chain relation
  • Proposition 2
  • proof
  • Lemma 1
  • proof
  • Lemma 2
  • proof
  • ...and 7 more