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

Abstract

We study the orbits of solutions to the Markoff-type equation $$X^2 + Y^2 + Z^2 = XYZ + AX + BY + CZ + D$$ in $\mathbb{F}_p,$ for fixed integers $A, B, C, D,$ under the symmetry group $\Gamma$ 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 defines the Relative Character Variety of the Four-Times Punctured Sphere, with $\Gamma$ arising from the Pure Mapping Class Group. Outside an explicit degeneracy locus, $\Gamma$ acts transitively on the bulk of solutions mod $p$ for density-one of primes, the remainder splitting into several small orbits reflecting finite orbits over $\mathbb{C}$. For the ``degenerate'' parameters, we show there are either two large orbits (most degenerate parameters) or four (the rest, excluding $(0, 0, 0, 4)$) for a density-one set of primes. These results are especially interesting for two subfamilies. The first, $$X^2 + Y^2 + Z^2 = XYZ + k,\,\,\,k\neq 4,$$ arises in the combinatorial group theory of $\text{SL}_2(\mathbb{F}_p)$; we very nearly prove the $Q$-classification conjecture of McCullough and Wanderley for density-one of primes. By work of Martin, this conjecture implies their Classification and $T$-Classification Conjectures. The second, $$x_1^2 + x_2^2 + x_3^2 + a_1x_2x_3 + a_2x_1x_3 + a_3x_1x_2 = (3+a_1+a_2+a_3)x_1x_2x_3,$$ arises from generalized cluster algebras. Our degeneracy notion specializes to that of de Courcy-Ireland, Litman, and Mizuno. For all nondegenerate and some degenerate surfaces in this subfamily, their results imply our orbit count (1, 2, or 4) holds for all sufficiently large primes.