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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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  • [1] I. Bernstein and A. Edmonds, On the classification of generic branched coverings of surfaces, Illinois J. Math. 28 (1984), 64-82. MR 730712 (85k:57004)
  • [2] S. A. Broughton, The homology and higher representations of the automorphism group of a Riemann surface, Trans. Amer. Math. Soc. 300 (1987), 153-158. MR 871669 (88m:30098)
  • [3] C. J. Earle, On the moduli of closed Riemann surfaces with symmetries, Advances in the Theory of Riemann Surfaces, Ann. of Math. Stud., no. 66, Princeton Univ. Press, Princeton, N.J., pp. 119-130. MR 0296282 (45:5343)
  • [4] H. Farkas and I. Kra, Riemann surfaces, Graduate Texts in Math., vol. 71, Springer-Verlag, Berlin and New York, 1980. MR 583745 (82c:30067)
  • [5] J. Gilman and D. Patterson, Intersection matrices for bases adapted to automorphisms of a compact Riemann surface, Riemann Surfaces and Related Topics, Ann. of Math. Stud., no. 97, Princeton Univ. Press, Princeton, N.J., 1981, pp. 149-166. MR 624812 (83a:30046)
  • [6] I. Guerrero, Holomorphic families of compact Riemann surfaces with automorphisms, Illinois J. Math. 26 (1982), 212-225. MR 650389 (83i:32031)
  • [7] W. J. Harvey, On branch loci in Teichmüller space, Trans. Amer. Math. Soc. 153 (1971), 387-399. MR 0297994 (45:7046)
  • [8] W. J. Harvey and C. Machlachlan, On mapping class groups and Teichmüller spaces, Proc. London Math. Soc. 30 (1975), 496-512. MR 0374414 (51:10614)
  • [9] S. Kerckhoff, The Nielsen realization problem, Ann. of Math. (2) 117 (1983), 235-265. MR 690845 (85e:32029)
  • [10] A. Kuribayashi, On analytic families of compact Riemann surfaces with non-trivial automorphism, Nagoya Math. J. 28 (1966), 119-165. MR 0217280 (36:371)
  • [11] I. Kuribayashi, On automorphism groups of a curve as linear groups, J. Math. Soc. Japan 39 (1987), 51-77. MR 867987 (88d:14020)
  • [12] A. M. Macbeath and D. Singerman, Spaces of subgroups and Teichmüller space, Proc. London Math. Soc. 31 (1975), 211-256. MR 0397022 (53:882)
  • [13] W. Magnus, A. Karass, and D. Solitar, Combinatorial group theory, Dover, New York, 1976. MR 0422434 (54:10423)
  • [14] H. Popp, Stratifikation von Quotientenmannigfaltigkeiten (in Characteristik 0) und insbesondere der Modulmannigfaltigkeiten für Kurven, J. Reine Angew. Math. 250 (1971), 12-41. MR 0301020 (46:180)
  • [15] J. F. X. Ries, The Prym variety for a cyclic unramified cover of a hyperelliptic surface, J. Reine Angew. Math. 340 (1983), 59-69. MR 691961 (85a:14020)
  • [16] -, The splitting of some Jacobi varieties using their automorphism groups, preprint.
  • [17] D. Singerman, Subgroups of Fuchsian groups and finite permutation groups, Bull. London Math. Soc. 2 (1970), 319-323. MR 0281805 (43:7519)
  • [18] -, Finitely maximal Fuchsian groups, J. London Math. Soc. 6 (1972), 29-38. MR 0322165 (48:529)

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