Modular companions in planar one-dimensional equisymmetric strata

TL;DR

This paper introduces a geometric measure to determine conformal equivalence among modular companions in the moduli space of Riemann surfaces with finite group actions, focusing on planar actions with four cone points.

Contribution

It constructs a moduli space for surfaces with specified group actions and develops a measure to compare modular companions' conformal structures.

Findings

Developed a geometric measure for conformal equivalence

Constructed a moduli space for surfaces with group actions

Applied methods to planar actions with four cone points

Abstract

Consider, in the moduli space of Riemann surfaces of a fixed genus, the subset of surfaces with non-trivial automorphisms. Of special interest are the numerous subsets of surfaces admitting an action of a given finite group, $G$, acting with a specific signature. In a previous study we declared two Riemann surfaces to be \emph{modular companions} if they have topologically equivalent $G$ actions, and that their $G$ quotients are conformally equivalent orbifolds. In this article we present a geometrically-inspired measure to decide whether two modular companions are conformally equivalent (or how different), respecting the $G$ action. Along the way, we construct a moduli space for surfaces with the specified $G$ action and associated equivariant tilings on these surfaces. We specifically apply the ideas to planar, finite group actions whose quotient orbifold is a sphere with four cone points.