The authors consider indecomposable degree \(n\) covers of Riemann surfaces with monodromy group an alternating or symmetric group of degree \(d\). They show that if the cover has five or more branch points then the genus grows rapidly with \(n\) unless either \(d = n\) or the curves have genus zero, there are precisely five branch points and \(n =d(d-1)/2\).

Similarly, if there is a totally ramified point, then without restriction on the number of branch points the genus grows rapidly with \(n\) unless either \(d=n\) or the curves have genus zero and \(n=d(d-1)/2\). One consequence of these results is that if \(f:X \rightarrow \mathbb{P}^1\) is indecomposable of degree \(n\) with \(X\) the generic Riemann surface of genus \(g \ge 4\), then the monodromy group is \(S_n\) or \(A_n\) (and both can occur for \(n\) sufficiently large).

The authors also show if that if \(f(x)\) is an indecomposable rational function of degree \(n\) branched at \(9\) or more points, then its monodromy group is \(A_n\) or \(S_n\). Finally, they answer a question of Elkies by showing that the curve parameterizing extensions of a number field given by an irreducible trinomial with Galois group \(H\) has large genus unless \(H=A_n\) or \(S_n\) or \(n\) is very small.