2025/02/05 by Broughton, S. Allen, Costa, Antonio F., Izquierdo, Milagros
#14H30 #14H37 #30F30 #Algebraic Geometry (math.AG) #Complex Variables (math.CV) #FOS: Mathematics #Geometric Topology (math.GT)
paper · doi:10.48550/arxiv.2502.03595
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 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.