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Tykhyy's Conjecture on finite mapping class group orbits

Samuel Bronstein, Arnaud Maret

TL;DR

The paper resolves the DT-driven facet of Tykhyy's Conjecture for SL2(C) representations of punctured spheres by proving there are no finite mapping class group orbits for DT representations when the sphere has at least seven punctures and identifying a unique 40-point jester's hat orbit for six punctures. It develops an inductive scheme using chained pants decompositions to reduce higher-puncture cases to established 4-punctured DT classifications, constraining action-angle coordinates via non-peripheral trace fields and a detailed Dehn-twist analysis. The work also recovers LT's 4-punctured results within the DT framework and completes the conjecture by leveraging the Corlette–Simpson alternative to link Zariski-dense finite-orbit representations to DT pullbacks or VHS structures, with additional consequences for Katz middle convolution and pullback families. Together, these results provide a complete classification of finite mapping class group orbits in genus-0 surface groups for DT and Zariski-dense cases and connect to broader isomonodromic and higher-genus dynamics.

Abstract

We classify the finite orbits of the mapping class group action on the character variety of Deroin--Tholozan representations of punctured spheres. In particular, we prove that the action has no finite orbits if the underlying sphere has 7 punctures or more. When the sphere has six punctures, we show that there is a unique 1-parameter family of finite orbits. Our methods also recover Tykhyy's classification of finite orbits for 5-punctured spheres. The proof is inductive and uses Lisovyy--Tykhyy's classification of finite mapping class group orbits for 4-punctured spheres as the base case for the induction. Our results on Deroin--Tholozan representations cover the last missing cases to complete the proof of Tykhyy's Conjecture on finite mapping class group orbits for $\mathrm{SL}_2\mathbb{C}$ representations of punctured spheres, after the recent work by Lam--Landesman--Litt.

Tykhyy's Conjecture on finite mapping class group orbits

TL;DR

The paper resolves the DT-driven facet of Tykhyy's Conjecture for SL2(C) representations of punctured spheres by proving there are no finite mapping class group orbits for DT representations when the sphere has at least seven punctures and identifying a unique 40-point jester's hat orbit for six punctures. It develops an inductive scheme using chained pants decompositions to reduce higher-puncture cases to established 4-punctured DT classifications, constraining action-angle coordinates via non-peripheral trace fields and a detailed Dehn-twist analysis. The work also recovers LT's 4-punctured results within the DT framework and completes the conjecture by leveraging the Corlette–Simpson alternative to link Zariski-dense finite-orbit representations to DT pullbacks or VHS structures, with additional consequences for Katz middle convolution and pullback families. Together, these results provide a complete classification of finite mapping class group orbits in genus-0 surface groups for DT and Zariski-dense cases and connect to broader isomonodromic and higher-genus dynamics.

Abstract

We classify the finite orbits of the mapping class group action on the character variety of Deroin--Tholozan representations of punctured spheres. In particular, we prove that the action has no finite orbits if the underlying sphere has 7 punctures or more. When the sphere has six punctures, we show that there is a unique 1-parameter family of finite orbits. Our methods also recover Tykhyy's classification of finite orbits for 5-punctured spheres. The proof is inductive and uses Lisovyy--Tykhyy's classification of finite mapping class group orbits for 4-punctured spheres as the base case for the induction. Our results on Deroin--Tholozan representations cover the last missing cases to complete the proof of Tykhyy's Conjecture on finite mapping class group orbits for representations of punctured spheres, after the recent work by Lam--Landesman--Litt.
Paper Structure (81 sections, 51 theorems, 67 equations, 9 tables)

This paper contains 81 sections, 51 theorems, 67 equations, 9 tables.

Table of Contents

  1. Introduction
  2. Results and overview
  3. Ideas of the proofs
  4. Motivations and related works
  5. Dynamics on character varieties
  6. Isomonodromy differential equations
  7. Higher genus
  8. Structure of the paper
  9. Acknowledgements
  10. Background
  11. Character varieties and mapping class group action
  12. Recap on DT representations
  13. Brief history
  14. Definition
  15. Triangle chains
  16. ...and 66 more sections

Key Result

Theorem A

For a punctured sphere $\Sigma$ with $7$ punctures or more, all the mapping class group orbits of conjugacy classes of DT representations are infinite.

Theorems & Definitions (126)

  • Theorem A: Theorem \ref{['thm:no-finite-orbit-for-n-geq-7']}
  • Theorem B: Theorems \ref{['thm:angle-vector-alpha-with-finite-orbits-n=6']} & \ref{['thm:existence-finite-orbit-n=6']}
  • Theorem C: Theorem \ref{['thm:tykhyy']}
  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Definition 2.4
  • Definition 2.5
  • proof
  • Definition 2.7
  • ...and 116 more