Markoff triples and Nielsen equivalence in $\text{SL}_2(\mathbb{F}_p)$

Abstract

In 2013, Darryl McCullough and Marcus Wanderley made a series of conjectures that describe the Nielsen equivalence classes and $T_2$-equivalence classes of pairs of generators for $\text{SL}_2(\mathbb{F}_q)$ and the Markoff equivalence classes of triples in $\mathbb{F}_q^3$ that solve $x^2+y^2+z^2=xyz+\kappa$ for some $\kappa\in\mathbb{F}_q$. (The case $\kappa=0$ was originally conjectured by Baragar in 1991.) We prove that one of the McCullough-Wanderley conjectures, the "Q-Classification Conjecture" on Markoff triples, implies the others. Then we prove that the Q-Classification Conjecture holds if $q=p$ is a prime such that $24{,}504{,}480$ does not divide $p^2-1$. More generally, for any integer $d$, we reduce the Q-Classification Conjecture for all primes $p\not\equiv \pm 1\,\text{mod}\,d$ to checking whether a roughly $2d\times 2d$ matrix with entries in $\mathbb{Q}[\kappa]$ is invertible. We (and SageMath) perform this invertibility check for all prime powers $d$ up to $17$, hence the modulus $24{,}504{,}480=2\text{lcm}(1,2,\dots,17)$.