Subvarieties of moduli space determined by finite groups acting on surfaces

Author:
John F. X. Ries

Journal:
Trans. Amer. Math. Soc. **335** (1993), 385-406

MSC:
Primary 14H15; Secondary 30F10, 30F20, 32G15

DOI:
https://doi.org/10.1090/S0002-9947-1993-1097170-2

MathSciNet review:
1097170

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Abstract: Suppose the finite group acts as orientation preserving homeomorphisms of the oriented surface of genus . This determines an irreducible subvariety of the moduli space of Riemann surfaces of genus consisting of all surfaces with a group of holomorphic homeomorphisms of the same topological type as . This family is determined by an equivalence class of epimorphisms from a Fuchsian group to whose kernel is isomorphic to the fundamental group of . To examine the singularity of along this family one needs to know the full automorphism group of a generic element of . In we show how to compute this from . Let denote the locus of all Riemann surfaces of genus whose automorphism group contains a subgroup isomorphic to . In we show that the irreducible components of this subvariety do not necessarily correspond to the families above, that is, the components cannot be put into a one-to-one correspondence with the topological actions of . In we examine the actions of on the spaces of holomorphic -differentials and on the first homology. We show that when is not cyclic, the characters of these actions do not necessarily determine the topological type of the action of on .

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DOI:
https://doi.org/10.1090/S0002-9947-1993-1097170-2

Article copyright:
© Copyright 1993
American Mathematical Society