Modular companions in planar one-dimensional equisymmetric strata

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 G , acting with a specific signature. In a previous study \cite{BrCoIz2}, we declared two Riemann surfaces to be modular companions if they have topologically equivalent G G actions, and that their G 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 G action. Along the way, we construct a moduli space for surfaces with the specified G 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.