# 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.

## Key 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.

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Source: https://tomesphere.com/paper/1907.07999