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

Let $k$ be an algebraically closed field of characteristic $p$ and let $X$ the projective line over with three points removed. We investigate which finite groups $G$ can arise as monodromy group \'{e}tale covers that are tamely ramified removed points. This provides new information about tame fundamental line. In particular, we show for each prime $p\ge 5$, there families symmetric $S_n$ or alternating $A_n$ infinitely many $n$. These come from moduli spaces elliptic curves $PSL_2(\mathbb{F}_\ell)$-structure, analysis uses work Bourgain, Gamburd, Sarnak, adapts Meiri Puder, Markoff triples modulo $\ell$.