Modular companions in planar one-dimensional equisymmetric strata
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| Format: | Preprint |
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2025
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| _version_ | 1866915141315985408 |
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| author | Broughton, S. Allen Costa, Antonio F. Izquierdo, Milagros |
| author_facet | Broughton, S. Allen Costa, Antonio F. Izquierdo, Milagros |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2502_03595 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Modular companions in planar one-dimensional equisymmetric strata Broughton, S. Allen Costa, Antonio F. Izquierdo, Milagros Geometric Topology Algebraic Geometry Complex Variables 14H30, 14H37, 30F30 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. |
| title | Modular companions in planar one-dimensional equisymmetric strata |
| topic | Geometric Topology Algebraic Geometry Complex Variables 14H30, 14H37, 30F30 |
| url | https://arxiv.org/abs/2502.03595 |