On branch loci in Teichmüller space
Author:
W. J. Harvey
Journal:
Trans. Amer. Math. Soc. 153 (1971), 387-399
MSC:
Primary 30A60; Secondary 58D15
DOI:
https://doi.org/10.1090/S0002-9947-1971-0297994-0
MathSciNet review:
0297994
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Abstract | References | Similar Articles | Additional Information
Abstract: The branch locus of the ramified covering of the space of moduli of Fuchsian groups with fixed presentation by the corresponding Teichmüller space is decomposed into a union of Teichmüller spaces, each characterised by a description of the action of the conformal self-mappings admitted by the underlying Riemann surfaces. Equivalence classes of subloci under the action of the modular group are studied, and counted in certain simple cases. One may compute as a result the number of conjugacy classes of elements of prime order in the mapping class group of closed surfaces.
- [1] L. Bers, On moduli of Riemann surfaces, Lecture Notes from Forschungsinstitut für Mathematik, Zurich, 1964.
- [2] Leon Greenberg, Maximal Fuchsian groups, Bull. Amer. Math. Soc. 69 (1963), 569–573. MR 148620, https://doi.org/10.1090/S0002-9904-1963-11001-0
- [3] W. J. Harvey, Cyclic groups of automorphisms of a compact Riemann surface, Quart. J. Math. Oxford Ser. (2) 17 (1966), 86–97. MR 201629, https://doi.org/10.1093/qmath/17.1.86
- [4] Saul Kravetz, On the geometry of Teichmüller spaces and the structure of their modular groups, Ann. Acad. Sci. Fenn. Ser. A I No. 278 (1959), 35. MR 0148906
- [5] Akikazu Kuribayashi, On analytic families of compact Riemann surfaces with non-trivial automorphisms, Nagoya Math. J. 28 (1966), 119–165. MR 217280
- [6] Joseph Lewittes, Automorphisms of compact Riemann surfaces, Amer. J. Math. 85 (1963), 734–752. MR 160893, https://doi.org/10.2307/2373117
- [7] E. K. Lloyd, Riemann surface transformation groups, Ph.D. Thesis, Birmingham, 1967.
- [8] A. M. Macbeath, Discontinuous groups, Proc. Summer School, Dundee, 1961.
- [9] A. M. Macbeath, Groups of homeomorphisms of a simply connected space, Ann. of Math. (2) 79 (1964), 473–488. MR 160848, https://doi.org/10.2307/1970405
- [10] A. M. Macbeath, The classification of non-euclidean plane crystallographic groups, Canadian J. Math. 19 (1967), 1192–1205. MR 220838, https://doi.org/10.4153/CJM-1967-108-5
- [11] Jakob Nielsen, Untersuchungen zur Topologie der geschlossenen zweiseitigen Flächen, Acta Math. 50 (1927), no. 1, 189–358 (German). MR 1555256, https://doi.org/10.1007/BF02421324
- [12] H. E. Rauch, A transcendental view of the space of algebraic Riemann surfaces, Bull. Amer. Math. Soc. 71 (1965), 1–39. MR 213543, https://doi.org/10.1090/S0002-9904-1965-11225-3
- [13] André Weil, On discrete subgroups of Lie groups, Ann. of Math. (2) 72 (1960), 369–384. MR 137792, https://doi.org/10.2307/1970140
- [14] Heiner Zieschang, Über Automorphismen ebener diskontinuierlicher Gruppen, Math. Ann. 166 (1966), 148–167 (German). MR 201521, https://doi.org/10.1007/BF01361444
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Additional Information
DOI:
https://doi.org/10.1090/S0002-9947-1971-0297994-0
Keywords:
Teichmüller spaces,
Fuchsian groups of first kind,
conformal automorphisms of Riemann surfaces,
mapping class group
Article copyright:
© Copyright 1971
American Mathematical Society