Markoff triples and generating pairs of $\mathrm{SL}_2(\mathbb{F}_p)$

Abstract

Consider the level sets of the Markoff equation $$\mathrm{M}_k: x^2 + y^2 + z^2 - xyz - 2 = k.$$ The phenomenon of strong approximation, as named by Bourgain, Gamburd, and Sarnak, predicts that every solution of $\mathrm{M}_k$ over $\mathbb{F}_p$ descends from a solution over $\mathbb{Z}$. Moreover, we expect that the action of Vieta involutions (taking $(x, y, z)$ to $(yz-x, y, z)$, $(x, xz-y, z)$, and $(x, y, xy-z)$) on $\mathrm{M}_k(\mathbb{F}_p)$ is essentially transitive. In terms of matrices, Vieta involutions correspond to Nielsen moves in pairs $(A, B) \in \mathrm{SL}_2(\mathbb{F}_p) \times \mathrm{SL}_2(\mathbb{F}_p)$ for which $\mathrm{tr}([A, B]) = k$. This correspondence is induced by \[\mathrm{Tr}: (A, B) \mapsto (\mathrm{tr}(A), \mathrm{tr}(B), \mathrm{tr}(AB)).\] McCullough and Wanderley conjectured that Nielsen moves connect two pairs $(A_1, B_1)$, $(A_2, B_2)$ of generators of $\mathrm{SL}_2(\mathbb{F}_p)$ if and only if $[A_1, B_1]$ is conjugate to $[A_2, B_2]$ or $[B_2, A_2]$. Based on this, one expects that generating pairs $(A, B)$ of $\mathrm{SL}_2(\mathbb{F}_p)$ for which $\mathrm{tr}([A, B]) = k$ determine a single orbit of $\mathrm{M}_k(\mathbb{F}_p)$, and the remaining exceptional orbits come from non-generating pairs of $\mathrm{SL}_2(\mathbb{F}_p)$. In this article, we describe the set of exceptional orbits of $\mathrm{M}_k(\mathbb{F}_p)$, showing that they agree with the finite orbits of the equation $\mathrm{M}_k$ over $\mathbb{C}$ found by Dubrovin and Mazzocco. Furthermore, we prove that the conjecture of McCullough and Wanderley is equivalent to strong approximation when $p \equiv 3 \mod{4}$. Lastly, we present the recent developments of Chen on the problem and use our classification of exceptional orbits to make a divisibility conjecture about the size of the largest orbit of $\mathrm{M}_k(\mathbb{F}_p)$.