Note on the Hurwitz-Nielsen realisation problem
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- by C. Maclachlan
- Proc. Amer. Math. Soc. 64 (1977), 87-90
- DOI: https://doi.org/10.1090/S0002-9939-1977-0442928-X
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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.References
- Joan S. Birman, Mapping class groups and their relationship to braid groups, Comm. Pure Appl. Math. 22 (1969), 213–238. MR 243519, DOI 10.1002/cpa.3160220206
- Joan S. Birman and Hugh 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 0292082
- Edward Fadell and Lee Neuwirth, Configuration spaces, Math. Scand. 10 (1962), 111–118. MR 141126, DOI 10.7146/math.scand.a-10517
- Saul Kravetz, On the geometry of Teichmüller spaces and the structure of their modular groups, Ann. Acad. Sci. Fenn. Ser. A I 278 (1959), 35. MR 0148906
- 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 0355113
- C. Maclachlan and W. J. Harvey, On mapping-class groups and Teichmüller spaces. 4, Proc. London Math. Soc. (3) 30 (1975), no. part, 496–512. MR 374414, DOI 10.1112/plms/s3-30.4.496
- Wilhelm Magnus, Über Automorphismen von Fundamentalgruppen berandeter Flächen, Math. Ann. 109 (1934), no. 1, 617–646 (German). MR 1512913, DOI 10.1007/BF01449158
- Jakob Nielsen, Abbildungsklassen endlicher Ordnung, Acta Math. 75 (1943), 23–115 (German). MR 13306, DOI 10.1007/BF02404101
- 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 (Italian). MR 29161 —, Bemaerkninger om endelige grupper af afbildningsklasser (Remarks on finite groups of mapping classes), Mat. Tidsskr. B 1950 (1950), 90-95. (Danish) MR 12, 349.
Bibliographic Information
- © Copyright 1977 American Mathematical Society
- 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