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.

arXiv commentExchanged 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