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Symmetric and Alternating Groups as Monodromy Groups of Riemann Surfaces I: Generic Covers and Covers with Many Branch Points: with an Appendix by R. Guralnick and R. Stafford
Robert M. Guralnick, University of Southern California, Los Angeles, CA, and John Shareshian, Washington University, St. Louis, MO

Memoirs of the American Mathematical Society
2007; 128 pp; softcover
Volume: 189
ISBN-10: 0-8218-3992-6
ISBN-13: 978-0-8218-3992-8
List Price: US$68
Individual Members: US$40.80
Institutional Members: US$54.40
Order Code: MEMO/189/886
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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.

Table of Contents

  • 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
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