Strong approximation for the Markoff equation

The Markoff equation is given by $x^2 + y^2 + z^2 - 3xyz = 0$. We say that the Markoff equation satisfies strong approximation at a prime $p$ if its integral points surject onto its $\mathbb{F}_p$-points. In 2016, Bourgain, Gamburd, and Sarnak conjectured that this holds at all primes $p$ and were able to establish their conjecture outside of a sparse but infinite set of primes. In this paper we establish a new result on the degree of a noncongruence modular curve over the $j$-line in terms of its moduli interpretation as a moduli space of elliptic curves with nonabelian level structures. Group-theoretically, the result amounts to a divisibility theorem on the cardinalities of Nielsen equivalence classes of generating pairs of finite groups. As a corollary, combined with results of Bourgain, Gamburd, and Sarnak, this establishes their conjecture for all but finitely many primes. Since their methods are effective, we have effectively reduced their conjecture to a finite computation.