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 $24,504,480=2\text{lcm}(1,2,\dots ,17)$.

62 pages. Typos from version 1 have been fixed, and further explanation has been added to certain proofs. Sage code for Algorithm 1 is on the author's webpage: dem6.people.clemson.edu