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arXiv · 2409.04379

Tykhyy's Conjecture on finite mapping class group orbits

Samuel Bronstein·Arnaud Maret
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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.

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Samuel Bronstein, Arnaud Maret. 2024-09-06. Tykhyy's Conjecture on finite mapping class group orbits. https://arxiv.org/abs/2409.04379

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