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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



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
MathSciNet review: 1097170
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Abstract: Suppose the finite group $ G$ acts as orientation preserving homeomorphisms of the oriented surface $ S$ of genus $ g$. This determines an irreducible subvariety $ \mathcal{M}_g^{[G]}$ of the moduli space $ {\mathcal{M}_g}$ of Riemann surfaces of genus $ g$ consisting of all surfaces with a group $ {G_1}$ of holomorphic homeomorphisms of the same topological type as $ G$. This family is determined by an equivalence class of epimorphisms $ \psi $ from a Fuchsian group $ \Gamma $ to $ G$ whose kernel is isomorphic to the fundamental group of $ S$. To examine the singularity of $ {\mathcal{M}_g}$ along this family one needs to know the full automorphism group of a generic element of $ \mathcal{M}_g^{[G]}$. In $ \S2$ we show how to compute this from $ \psi $. Let $ \mathcal{M}_g^G$ denote the locus of all Riemann surfaces of genus $ g$ whose automorphism group contains a subgroup isomorphic to $ G$. In $ \S3$ 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 $ G$. In $ \S4$ we examine the actions of $ G$ on the spaces of holomorphic $ k$-differentials and on the first homology. We show that when $ G$ is not cyclic, the characters of these actions do not necessarily determine the topological type of the action of $ G$ on $ S$.

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Article copyright: © Copyright 1993 American Mathematical Society