Character varieties on a four-holed sphere

Abstract

For each $\mathbf{k}\in\mathbb{C}^4$, let $V_\mathbf{k}$ be the character variety on a four-holed sphere and $Γ_\mathbf{k}$ the group generated by the Vieta involution maps. First, under a certain condition, we find a fundamental domain for $Γ_\mathbf{k}$-action on $V_\mathbf{k}$, expressed via inequalities. Second, we show that it is decidable whether or not two integral solutions for $V_\mathbf{k}$ are in the same $Γ_\mathbf{k}$-orbit and in the same mapping class group orbit. To achieve our goals, we introduce graphs corresponding to the $Γ_\mathbf{k}$-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.