Tykhyy's Conjecture on finite mapping class group orbits
Fuente:
arXiv
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| Autori principali: | , |
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| _version_ | 1866908720495067136 |
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| author | Bronstein, Samuel Maret, Arnaud |
| author_facet | Bronstein, Samuel Maret, Arnaud |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2409_04379 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Tykhyy's Conjecture on finite mapping class group orbits Bronstein, Samuel Maret, Arnaud Dynamical Systems Algebraic Geometry Geometric Topology 37C25, 57K20, 20C15, 34M56 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. |
| title | Tykhyy's Conjecture on finite mapping class group orbits |
| topic | Dynamical Systems Algebraic Geometry Geometric Topology 37C25, 57K20, 20C15, 34M56 |
| url | https://arxiv.org/abs/2409.04379 |