Strong Approximation for the Character Variety of the Four-Times Punctured Sphere

Abstract

We study the orbits of the solutions to the Markoff-type equation $$X^2 + Y^2 + Z^2 = XYZ +AX + BY + CZ + D$$ in $\mathbb{F}_p$ for fixed integers $A, B, C,$ and $D$ under the group of symmetries $\Gamma$ generated by \[\begin{split}&V_1: (x, y, z)\mapsto (A + yz - x, y, z),\\ &V_2: (x, y, z)\mapsto (x, B + xz - y, z),\text{ and}\\ &V_3: (x, y, z)\mapsto (x, y, C + xy - z).\end{split}\] This equation arises as the Relative Character Variety of the Four-Times Punctured Sphere, and $\Gamma$ arises from the Pure Mapping Class Group. For most parameters we show that there is a density one set of primes $p$ such that $\Gamma$ acts transitively on the bulk of the solutions mod $p$, with the remainder breaking up into a few small orbits arising from finite orbits within the solutions over $\mathbb{C}$. We classify those ``degenerate'' parameters to which this result does not apply, and show there are either 2 (for most degenerate parameters) or 4 (for the remaining degenerate parameters other than $(0, 0, 0, 4)$) large orbits modulo density one of primes. Our results become especially interesting when applied to two special subfamilies. The first is $$X^2 + Y^2 + Z^2 = XYZ + k$$ for $k \neq 4$, which arises in the study of the combinatorial group theory of $\text{SL}_2(\mathbb{F}_p)$. Our results very nearly prove the $Q$-classification conjecture of McCullough and Wanderley for density 1 of primes. The second subfamily is $$x_1^2 + x_2^2 + x_3^2 + a_1x_2x_3 + a_2x_1x_3 + a_3x_1x_2 = (3+a_1+a_2+a_3)x_1x_2x_3,$$ which arises from certain generalized cluster algebras. Here, our notion of degenerate parameters specializes to the degeneracy condition of de Courcy-Ireland, Litman, and Mizuno. For all nondegenerate and some degenerate surfaces in this family, their results imply that our count of large orbits (1, 2, or 4) applies to all sufficiently large primes $p$.