It is a classical problem to classify the connected components of the Hurwitz moduli spaces H(g,n) classifying finite covers of genus g curves with n branch points. Over a century ago, Clebsch and Hurwitz established connectivity for the subspace classifying simply branched covers of the projective line, which led to the first proof of the irreducibility of the moduli space of curves of a given genus. More recently, the work of Dunfield-Thurston and Conway-Parker establish connectivity in certain situations where the monodromy group G is fixed and either g or n are allowed to be large. When (g,n) are fixed and the monodromy group G is allowed to vary, far less is known. In this talk we will prove a connectivity result of this type -- namely, for large enough primes p, we will establish the connectivity of the moduli space of SL(2,p)-covers of elliptic curves, only branched over the origin with ramification indices 2p. The proof exploits a connection with Diophantine aspects of the Markoff equation x^2 + y^2 + z^2 - xyz = 0, and combines asymptotic results of Bourgain, Gamburd, and Sarnak with a rigidity property coming from algebraic geometry. As an application we resolve a conjecture of Bourgain, Gamburd, and Sarnak, and a question of Frobenius from 1913.