  Let $k$ be an algebraically closed field of characteristic $p$ and let $X$
the projective line over $k$ with three points removed. We investigate which
finite groups $G$ can arise as the monodromy group of finite \'{e}tale 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 $p\ge 5$, 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 $PSL_2(\mathbb{F}_\ell)$-structure, and the analysis uses
work of Bourgain, Gamburd, and Sarnak, and adapts work of Meiri and Puder,
about Markoff triples modulo $\ell$.
