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, $$X^2+Y^2+Z^2=XYZ+k,k\ne 4,$$ 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, $$x_1^2+x_2^2+x_3^2+a_1{x}_2{x}_3+a_2{x}_1{x}_3+a_3{x}_1{x}_2=(3+a_1+a_2+a_3)x_1{x}_2{x}_3,$$ 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.

Exchanged Sections 6 and 7. Major corrections to and simplifications of the proof of Theorem 7.1 (according to the new section numbering), fixed a gap in the proof of Proposition 9.1, re-organized and lemmatized the proof of Theorem 11.1, and made many minor changes to correct typos and improve exposition. 72 pages, 4 figures