Abstract
A compact Riemann surface X is called a (p, n)-gonal surface if there exists a group of automorphisms C of X (called a (p, n)-gonal group) of prime order p such that the orbit space X/C has genus n. We derive some basic properties of (p, n)-gonal surfaces considered as generalizations of hyperelliptic surfaces and also examine certain properties which do not generalize. In particular, we find a condition which guarantees all (p, n)-gonal groups are conjugate in the full automorphism group of a (p, n)-gonal surface, and we find an upper bound for the size of the corresponding conjugacy class. Furthermore we give an upper bound for the number of conjugacy classes of (p, n)-gonal groups of a (p, n)-gonal surface in the general case. We finish by analyzing certain properties of quasiplatonic (p, n)-gonal surfaces. An open problem and two conjectures are formulated in the paper.
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Grzegorz Gromadzki was supported by the Research Grant N N201 366436 of the Polish Ministry of Sciences and Higher Education.
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Gromadzki, G., Weaver, A. & Wootton, A. On gonality of Riemann surfaces. Geom Dedicata 149, 1–14 (2010). https://doi.org/10.1007/s10711-010-9459-x
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DOI: https://doi.org/10.1007/s10711-010-9459-x
Keywords
- Automorphism groups of Riemann surfaces
- Hyperelliptic Riemann surfaces
- p-Hyperelliptic Riemann surfaces
- p-Gonal Riemann surfaces
- Fuchsian groups
- Uniformization theorem