Symmetric and alternating groups as monodromy groups of Riemann surfaces. I. Generic covers and covers with many branch points

Abstract

Introduction and statement of main results Notation and basic lemmas Examples Proving the main results on five or more branch points--Theorems 1.1.1 and 1.1.2 Actions on $2$-sets--the proof of Theorem 4.0.30 Actions on $3$-sets--the proof of Theorem 4.0.31 Nine or more branch points--the 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 $n$-cycle--the proof of Theorem 1.2.1 Galois groups of trinomials--the proofs of Propositions 1.4.1 and 1.4.2 and Theorem 1.4.3 Appendix A. Finding small genus examples by computer search--by R. Guralnick and R. Stafford Appendix. Bibliography.