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Note on the Hurwitz-Nielsen realisation problem


Author: C. Maclachlan
Journal: Proc. Amer. Math. Soc. 64 (1977), 87-90
MSC: Primary 55E99; Secondary 57E05, 32G15, 30A46
DOI: https://doi.org/10.1090/S0002-9939-1977-0442928-X
MathSciNet review: 0442928
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Abstract: This problem, which is equivalent to the problem of showing that every finite subgroup of the Teichmüller modular group of a compact surface of genus g has a fixed point in the corresponding Teichmüller space, is solved for $ g = 2$. The generalised problem for n-punctured surfaces of genus $ g \geqslant 1$ and $ 1 \leqslant n \leqslant 4$ is also solved.


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  • [1] J. S. Birman, Mapping class groups and their relationship to braid groups, Comm. Pure Appl. Math. 22 (1969), 213-238. MR 39 #4840. MR 0243519 (39:4840)
  • [2] J.S. Birman and H.M. Hilden, On the mapping class groups of closed surfaces as covering spaces, Advances in the Theory of Riemann Surfaces (Proc. Conf., Stony Brook, N.Y., 1969), Ann. of Math. Studies, No. 66, Princeton Univ. Press, Princeton, N.J., 1971, pp. 81-115. MR 45#1169. MR 0292082 (45:1169)
  • [3] E. Fadell and L. Neuwirth, Configuration spaces, Math. Scand. 10 (1962), 111-118. MR 25 #4537. MR 0141126 (25:4537)
  • [4] S. Kravetz, On the geometry of Teichmüller spaces and the structure of their modular groups, Ann. Acad. Sci. Fenn. Ser. A I, No. 278 (1959), 1-35. MR 26 #6402. MR 0148906 (26:6402)
  • [5] C. Maclachlan, Modular groups and fiber spaces over Teichmüller spaces, Discontinuous Groups and Riemann Surfaces (Proc. Conf., Univ. Maryland, College Park, Md., 1973), Ann. of Math. Studies, No. 79, Princeton Univ. Press, Princeton, N. J., 1974, pp. 297-314. MR 50 #7590. MR 0355113 (50:7590)
  • [6] C. Maclachlan and W. J. Harvey, On mapping-class groups and Teichmüller spaces, Proc. London Math. Soc. (3) 30 (1975), 496-512. MR 51 #10614. MR 0374414 (51:10614)
  • [7] W. Magnus, Über Automorphismen von Fundamentalgruppen Berandeter Flächen, Math. Ann. 109 (1934), 617-646. MR 1512913
  • [8] J. Nielsen, Abbildungsklassen endlicher Ordnung, Acta Math. 75 (1943), 23-115. MR 7, 137. MR 0013306 (7:137a)
  • [9] W. Fenchel, Estensioni di gruppi discontinui e trasformazioni periodiche delle superficie, Atti Accad. Naz. Lincei Rend. Cl. Sci. Fis. Mat. Nat. (8) 5 (1948), 326-329. MR 10, 558. MR 0029161 (10:558c)
  • [10] -, Bemaerkninger om endelige grupper af afbildningsklasser (Remarks on finite groups of mapping classes), Mat. Tidsskr. B 1950 (1950), 90-95. (Danish) MR 12, 349.

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Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1977-0442928-X
Keywords: Teichmüller space, Teichmüller modular group, Fuchsian group
Article copyright: © Copyright 1977 American Mathematical Society

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