Omnipersistent signatures

TL;DR

This paper introduces a new arithmetic-based approach to classifying group signatures on Riemann surfaces, identifying signatures that appear across all genera, simplifying the classification process.

Contribution

It provides a comprehensive list of signatures that can occur in any genus and those that always appear as signatures of group actions, laying foundational groundwork.

Findings

List of signatures possible in all genera

Subset of signatures that always correspond to group actions

Simplification of classification process

Abstract

In this note, we lay the groundwork for a new approach to the problem of group-signature classification of group actions on closed Riemann surfaces. This new approach first focuses on analyzing the low level arithmetic conditions on signatures before invoking the more complicated group theory. We provide the complete first step in this approach by giving the complete list of signatures which arithmetically could appear as a signature in every possible genus, and the subset of those which do appear as the signature of a group action in every possible genus.