  In this paper we establish a congruence on the degree of the map from a
component of a Hurwitz space of covers of elliptic curves to the moduli stack
of elliptic curves. Combinatorially, this can be expressed as a congruence on
the cardinalities of Nielsen equivalence classes of generating pairs of finite
groups. Building on the work of Bourgain, Gamburd, and Sarnak, we apply this
congruence to show that for all but finitely many primes $p$, the group of
Markoff automorphisms acts transitively on the nonzero $\mathbb{F}_p$-points of
the Markoff equation $x^2 + y^2 + z^2 - 3xyz = 0$. This yields a strong
approximation property for the Markoff equation, the finiteness of congruence
conditions satisfied by Markoff numbers, and the connectivity of a certain
infinite family of Hurwitz spaces of $\text{SL}_2(\mathbb{F}_p)$-covers of
elliptic curves. With possibly finitely many exceptions, this resolves a
conjecture of Bourgain, Gamburd, and Sarnak, first posed by Baragar in 1991.
Since their methods are effective, this reduces the conjecture to a finite
computation.

Number Theory;Algebraic Geometry