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Modular companions in planar one-dimensional equisymmetric strata

S. Allen Broughton, Antonio F. Costa, Milagros Izquierdo

TL;DR

The paper develops a unified framework for the moduli of finite-group actions on Riemann surfaces by combining horizontal invariants (quotients as orbifolds) with vertical invariants (monodromies), encapsulated in the moduli spaces $ ext{S}_{G, rak s}=( ext{T}_{ rak s} imes ext{F}_{ rak s})/M_{ rak s}$. It introduces modular companions as connected components over a fixed quotient, and then builds a geometrical toolkit—equivariant tilings, dual/Cayley graphs, and partial isometries—to distinguish these companions when the quotients coincide but the actions are not conformally equivalent. The core technical advancement is the explicit, computable machinery (cut systems, crossover transformations, sector labelling, and tiling data) that allows one to detect edge collapses, multiple intersections, and vertex collapses, thereby differentiating modular companions in planar, four-cone-point strata. The methods yield both qualitative and quantitative insights (including explicit tables and partial isometry matrices) into how modular companions relate within the equisymmetric stratification and how they can be distinguished or connected by deformations. Together, these results provide new, geometry-driven tools for understanding automorphism-type stratifications in low-dimensional moduli spaces with concrete computational pathways. All mathematical constructs are handled in the orbifold/covering-space framework, enabling precise comparisons of actions via horizontal/vertical data and combinatorial tiling structures.

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.

Modular companions in planar one-dimensional equisymmetric strata

TL;DR

The paper develops a unified framework for the moduli of finite-group actions on Riemann surfaces by combining horizontal invariants (quotients as orbifolds) with vertical invariants (monodromies), encapsulated in the moduli spaces . It introduces modular companions as connected components over a fixed quotient, and then builds a geometrical toolkit—equivariant tilings, dual/Cayley graphs, and partial isometries—to distinguish these companions when the quotients coincide but the actions are not conformally equivalent. The core technical advancement is the explicit, computable machinery (cut systems, crossover transformations, sector labelling, and tiling data) that allows one to detect edge collapses, multiple intersections, and vertex collapses, thereby differentiating modular companions in planar, four-cone-point strata. The methods yield both qualitative and quantitative insights (including explicit tables and partial isometry matrices) into how modular companions relate within the equisymmetric stratification and how they can be distinguished or connected by deformations. Together, these results provide new, geometry-driven tools for understanding automorphism-type stratifications in low-dimensional moduli spaces with concrete computational pathways. All mathematical constructs are handled in the orbifold/covering-space framework, enabling precise comparisons of actions via horizontal/vertical data and combinatorial tiling structures.

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.

Paper Structure

This paper contains 58 sections, 13 theorems, 65 equations, 5 figures, 1 algorithm.

Table of Contents

  1. Overview, motivation, and background
  2. Overview and motivation
  3. Goal 1.
  4. Goal 2.
  5. Motivation
  6. Examples and computation
  7. Outline
  8. Acknowledgement
  9. Background
  10. Orbifolds
  11. Equisymmetry
  12. Group actions and signatures
  13. Uniformization
  14. Monodromies and path lifting
  15. Regular covering spaces and path lifting
  16. ...and 43 more sections

Key Result

Proposition 1.1

Suppose that $(X, \mathcal{G})$ is an orbifold pair, $X$ is connected, and $\mathcal{F}$ is a finite set upon which $\mathcal{G}$ acts via $\left(g,v\right) \rightarrow g\cdot v=gv$. Let $\mathcal{G}$ act upon $X\times\mathcal{F}$ by the diagonal action $g\cdot (x,v)=(g\cdot x,g\cdot v)$, and consi Then, we have the following.

Figures (5)

  • Figure 1: Ribbon graph for $\mathcal{E}_1$, with orientation markings
  • Figure 2: Various cut systems $\mathcal{E}_i$ on the sphere
  • Figure 3: Polygons $\mathcal{P}_i$ corresponding to $\mathcal{E}_i$
  • Figure 4: Crossover loops (red) for $\mathcal{E}_1$ with orientation markings
  • Figure 5: Crossover loops (red) for $\mathcal{E}_2$ with orientation markings

Theorems & Definitions (41)

  • Proposition 1.1
  • Remark 1.2
  • Proposition 1.3
  • Remark 1.4
  • Example 1.5
  • Definition 2.2
  • Remark 2.3
  • Definition 2.4
  • Remark 2.5
  • Remark 2.6
  • ...and 31 more