Memoirs of the American Mathematical Society 2007; 128 pp; softcover Volume: 189 ISBN10: 0821839926 ISBN13: 9780821839928 List Price: US$68 Individual Members: US$40.80 Institutional Members: US$54.40 Order Code: MEMO/189/886
 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 \(n =d(d1)/2\). 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 \(n=d(d1)/2\). 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. Table of Contents  Introduction and statement of main results
 Notation and basic lemmas
 Examples
 Proving the main results on five or more branch pointsTheorems 1.1.1 and 1.1.2
 Actions on \(2\)setsthe proof of Theorem 4.0.30
 Actions on \(3\)setsthe proof of Theorem 4.0.31
 Nine or more branch pointsthe proof of Theorem 4.0.34
 Actions on cosets of some \(2\)homogeneous and \(3\)homogeneous groups
 Actions on \(3\)sets compared to actions on larger sets
 A transposition and an \(n\)cycle
 Asymptotic behavior of \(g_k(E)\)
 An \(n\)cyclethe proof of Theorem 1.2.1
 Galois groups of trinomialsthe proofs of Propositions 1.4.1 and 1.4.2 and Theorem 1.4.3
 Appendix A. Finding small genus examples by computer searchby R. Guralnick and R. Stafford
 Appendix. Bibliography
