A Lower Bound for the Number of Group Actions on a Compact Riemann Surface

TL;DR

This paper establishes a quadratic lower bound on the number of distinct group actions on compact Riemann surfaces of fixed genus by introducing and analyzing a new coarse signature space called skeletal signatures.

Contribution

It introduces the skeletal signature space $\\mathcal{K}_\sigma$ and provides a conjectural description, advancing understanding of group actions on Riemann surfaces.

Findings

Number of group actions grows at least quadratically with genus.

Introduces the skeletal signature space $\mathcal{K}_\sigma$ for classifying actions.

Provides properties and conjectural framework for $\mathcal{K}_\sigma$.

Abstract

We prove that the number of distinct group actions on compact Riemann surfaces of a fixed genus $\sigma \geq 2$ is at least quadratic in $\sigma$. We do this through the introduction of a coarse signature space, the space $\mathcal{K}_\sigma$ of {\em skeletal signatures} of group actions on compact Riemann surfaces of genus $\sigma$. We discuss the basic properties of $\mathcal{K}_\sigma$ and present a full conjectural description.