Non-Abelian Simple Groups Act with Almost All Signatures

TL;DR

This paper characterizes when certain arithmetic conditions on signatures guarantee a group action on Riemann surfaces, showing that all non-Abelian finite simple groups satisfy these conditions for almost all signatures.

Contribution

It provides necessary and sufficient conditions on groups for the arithmetic signature conditions to be sufficient, highlighting a key property of non-Abelian finite simple groups.

Findings

Non-Abelian finite simple groups satisfy the signature conditions for almost all tuples.

Derived criteria are both necessary and sufficient for group actions on Riemann surfaces.

The results connect group theory with topological data encoding in surface actions.

Abstract

The topological data of a group action on a compact Riemann surface is often encoded using a tuple $(h;m_1,\dots ,m_s)$ called its signature. There are two easily verifiable arithmetic conditions on a tuple necessary for it to be a signature of some group action. In the following, we derive necessary and sufficient conditions on a group $G$ for when these arithmetic conditions are in fact sufficient to be a signature for all but finitely many tuples that satisfy them. As a consequence, we show that all non-Abelian finite simple groups exhibit this property.