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Signatures of alternating group actions with non-zero quotient genus

Jennifer Paulhus, Aaron Wootton

Abstract

We classify up to signature all the ways the alternating group $A_n$ can act on a compact Riemann surfaces when the quotient genus is greater than $0$. In particular, we prove that for $A_n$ with $n>6$ every potential signature for the group acting with quotient genus greater than $0$ is an actual signature. We also show that in the case of $n=5$, respectively $n=6$, the only failure is for $[1;2]$, respectively $[1;3]$. Along the way we also prove that for any finite simple non-abelian group, all potential signatures with quotient genus greater than $1$ are actual signatures.

Signatures of alternating group actions with non-zero quotient genus

Abstract

We classify up to signature all the ways the alternating group can act on a compact Riemann surfaces when the quotient genus is greater than . In particular, we prove that for with every potential signature for the group acting with quotient genus greater than is an actual signature. We also show that in the case of , respectively , the only failure is for , respectively . Along the way we also prove that for any finite simple non-abelian group, all potential signatures with quotient genus greater than are actual signatures.

Paper Structure

This paper contains 9 sections, 19 theorems, 15 equations.

Table of Contents

  1. Introduction
  2. Proof Outline and Core Results
  3. Overview of Proof
  4. Core Results
  5. Building Elements in $A_n$
  6. The case when periods are divisible by small primes
  7. The case $[1;n_1, n_2, \ldots, n_r]$ for $r\geq 2$
  8. The case $[1;k]$
  9. Computations for small $n$

Key Result

Theorem 1.1

A finite group $G$ acts on a compact Riemann surface $S$ of genus $\sigma \geq 2$ with signature $[h; n_1,\ldots, n_r]$ if and only if:

Theorems & Definitions (30)

  • Theorem 1.1: Riemann's Existence Theorem
  • Theorem 1.2
  • proof
  • Theorem 1.3
  • Proposition 2.1
  • Theorem 2.2: Bertram, Theorem 2, Bertram
  • Theorem 2.3: Xu, Lemma, pg. 339 Xu
  • Theorem 2.4: Miller pg. 25, Miller
  • Theorem 2.5: Jones, Corollary 1.3 Jones
  • Definition 2.1
  • ...and 20 more