On Riemann Surfaces of genus g with 4g automorphisms

TL;DR

This paper classifies Riemann surfaces of genus g with 4g automorphisms, revealing their structure in the moduli space and identifying real surfaces and their topological properties.

Contribution

It determines all such surfaces for all g ≥ 2 and describes their placement within the moduli space, including real structures and boundary behavior.

Findings

Surfaces form a real Riemann surface in the moduli space for most g

The set of real surfaces consists of three intervals

Closure in the Deligne-Mumford compactification is a closed Jordan curve

Abstract

We determine, for all genus $g\geq2$ the Riemann surfaces of genus $g$ with $4g$ automorphisms. For $g\neq$ $3,6,12,15$ or $30$, this surfaces form a real Riemann surface $\mathcal{F}_{g}$ in the moduli space $\mathcal{M}_{g}$: the Riemann sphere with three punctures. The set of real Riemann surfaces in $\mathcal{F}_{g}$ consists of three intervals its closure in the Deligne-Mumford compactification of $\mathcal{M}_{g}$ is a closed Jordan curve.