Modular companions in planar one-dimensional equisymmetric strata
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, , acting with a specific signature. In a previous study we declared two Riemann surfaces to be \emph{modular companions} if they have topologically equivalent actions, and that their 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 action. Along the way, we construct a moduli space for surfaces with the specified 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.
Cite
@article{arxiv.2502.03595,
title = {Modular companions in planar one-dimensional equisymmetric strata},
author = {S. Allen Broughton and Antonio F. Costa and Milagros Izquierdo},
journal= {arXiv preprint arXiv:2502.03595},
year = {2025}
}