Tamely ramified covers of the projective line with alternating and symmetric monodromy

Let k be an algebraically closed field of characteristic p and X the projective line over k with three points removed. We investigate which finite groups G can arise as the monodromy group of finite étale covers of X that are tamely ramified over the three removed points. This provides new information about the tame fundamental group of the projective line. In particular, we show that for each prime p ≥ 5, there are families of tamely ramified covers with monodromy the symmetric group S_n or alternating group A_n for infinitely many n. These covers come from the moduli spaces of elliptic curves with PSL₂ (_ℓ )-structure, and the analysis uses work of Bourgain, Gamburd, and Sarnak, and adapts work of Meiri and Puder about Markoff triples modulo ℓ.