# Symmetric and Alternating Groups as Monodromy Groups of Riemann Surfaces I: Generic Covers and Covers with Many Branch Points

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*Robert M. Guralnick; John Shareshian*

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

# Table of Contents

## Symmetric and Alternating Groups as Monodromy Groups of Riemann Surfaces I: Generic Covers and Covers with Many Branch Points

- Contents v6 free
- Chapter 1. Introduction and statement of main results 18 free
- Chapter 2. Notation and basic lemmas 1320
- Chapter 3. Examples 2128
- Chapter 4. Proving the main results on five or more branch points - Theorems 1.1.1 and 1.1.2 2936
- Chapter 5. Actions on 2-sets - the proof of Theorem 4.0.30 3340
- Chapter 6. Actions on 3-sets - the proof of Theorem 4.0.31 6572
- Chapter 7. Nine or more branch points - the proof of Theorem 4.0.34 7986
- Chapter 8. Actions on cosets of some 2-homogeneous and 3-homogeneous groups 8188
- Chapter 9. Actions on 3-sets compared to actions on larger sets 8996
- Chapter 10. A transposition and an n-cycle 97104
- Chapter 11. Asymptotic behavior of g[sub(k)] (E) 103110
- Chapter 12. An n-cycle - the proof of Theorem 1.2.1 107114
- Chapter 13. Galois groups of trinomials - the proofs of Propositions 1.4.1 and 1.4.2 and Theorem 1.4.3 117124
- Appendix A. Finding small genus examples by computer search 123130
- Bibliography 127134