Tykhyy's Conjecture on finite mapping class group orbits

TL;DR

This paper classifies finite orbits of the mapping class group on character varieties of punctured spheres, proving no finite orbits for spheres with 7 or more punctures and identifying a unique family for six punctures, completing Tykhyy's conjecture.

Contribution

It provides a complete classification of finite orbits for Deroin--Tholozan representations on punctured spheres, confirming Tykhyy's conjecture for all cases.

Findings

No finite orbits for spheres with 7 or more punctures.

Unique 1-parameter family of finite orbits for 6-punctured spheres.

Recovery of Tykhyy's classification for 5-punctured spheres.

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.