Automorphism groups of compact Riemann surfaces of genus five. (English) Zbl 0709.30037
Let X be a compact Riemann surface of genus \(\geq 2\), and let AG be a group of automorphisms of X. Since AG acts on the vector space of holomorphic differentials on X, it can be represented as a subgroup R(X,AG) of \(GL_ g({\mathbb{C}})\). It arises a problem: Determine all subgroups of \(GL_ g({\mathbb{C}})\) which are conjugate to R(X,AG) for some X and some AG. For \(g=2\), this problem was already solved by I. Kuribayashi and for \(g=3,4\) it was solved by I. Kuribayashi and the first author. In this paper, the authors solve it for \(g=5\).
Reviewer: T.Kato
MSC:
| 30F10 | Compact Riemann surfaces and uniformization |
| 30C20 | Conformal mappings of special domains |
| 32M05 | Complex Lie groups, group actions on complex spaces |
References:
| [1] | Harvey, W. J., Cyclic groups of automorphisms of a compact Riemann surface, Quart. J. Math. Oxford Ser., 17, 2, 86-97 (1966) · Zbl 0156.08901 |
| [2] | Harvey, W. J., On branch loci in Teichmüller space, Trans. Amer. Math. Soc., 153, 387-399 (1971) · Zbl 0211.10503 |
| [3] | Guerrero, I., Holomorphic families of compact Riemann surfaces with automorphisms, Illinois J. Math., 26, 212-225 (1984) · Zbl 0504.32018 |
| [4] | Kuribayashi, A.; Murase, N., Some applications of the existence theorem and the representability theorem of a Riemann surface, Bull. Fac. Sci. Engrg. Chuo Univ., 27, 1-53 (1984) · Zbl 0574.14025 |
| [5] | Kuribayashi, I., On certain curves of genus three with many automorphisms, Tsukuba J. Math., 6, 271-288 (1982) · Zbl 0534.14017 |
| [6] | Kuribayashi, I., On an algebraization of the Riemann-Hurwitz relation, Kodai Math. J., 7, 222-237 (1984) · Zbl 0575.32025 |
| [7] | Kuribayashi, I., Classification of automorphism groups of compact Riemann surfaces of genus two (1986), preprint, Tsukuba · Zbl 0589.30045 |
| [8] | Kuribayashi, I., On automorphism groups of a curve as linear groups, J. Math. Soc. Japan, 29, 51-77 (1987) · Zbl 0611.14027 |
| [10] | Maclachlan, C., A bound for the number of automorphisms of a compact Riemann, surface, J. London Math. Soc., 44, 265-272 (1969) · Zbl 0164.37904 |
| [11] | Momose, F., Galois coverings of curves with given representations on differential forms (1986), preprint, Tsukuba |
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