Character varieties on a four-holed sphere
Abstract
For each k ∈ C 4 \mathbf {k}\in \mathbb {C}^4 , let V k V_\mathbf {k} be the character variety on a four-holed sphere and Γ k \Gamma _\mathbf {k} the group generated by the Vieta involution maps. First, under a certain condition, we find a fundamental domain for Γ k \Gamma _\mathbf {k} -action on V k V_\mathbf {k} , expressed via inequalities. Second, we show that it is decidable whether or not two integral solutions for V k V_\mathbf {k} are in the same Γ k \Gamma _\mathbf {k} -orbit and in the same mapping class group orbit. To achieve our goals, we introduce graphs corresponding to the Γ k \Gamma _\mathbf {k} -orbits and the mapping class group orbits, and classify their restricted global shapes by analyzing the limited local edge configurations at each vertex, using a descent argument.
Fields of Study & Subjects
Advanced Algebra and Geometry
Algebraic Geometry and Number Theory
Finite Group Theory Research
Publication details
- Journal / source
- Proceedings of the American Mathematical Society
- Publisher
- American Mathematical Society (AMS)
- Volume
- 154
- Issue
- 10
- Pages
- 4091–4106
- Publication date
- 13 May 2026
- Language
- EN