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On branch loci in Teichmüller space
Author:
W. J. Harvey
Journal:
Trans. Amer. Math. Soc. 153 (1971), 387-399
MSC:
Primary 30A60; Secondary 58D15
MathSciNet review:
0297994
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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.
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- [1]
- L. Bers, On moduli of Riemann surfaces, Lecture Notes from Forschungsinstitut für Mathematik, Zurich, 1964.
- [2]
- L. Greenberg, Maximal Fuchsian groups, Bull. Amer. Math. Soc. 69 (1963), 569-573. MR 26 #6127. MR 0148620 (26:6127)
- [3]
- W. J. Harvey, Cyclic groups of automorphisms of a compact Riemann surface, Quart. J. Math. Oxford Ser. (2) 17 (1966), 86-97. MR 34 #1511. MR 0201629 (34:1511)
- [4]
- S. Kravetz, On the geometry of Teichmüller spaces and the structure of their modular groups, Ann. Acad. Sci. Fenn. Ser. AI No. 278 (1959). MR 26 #6402. MR 0148906 (26:6402)
- [5]
- A. Kuribayashi, On analytic families of compact Riemann surfaces with non-trivial automorphisms, Nagoya Math. J. 28 (1966), 119-165. MR 36 #371. MR 0217280 (36:371)
- [6]
- J. Lewittes, Automorphisms of compact Riemann surfaces, Amer. J. Math. 85 (1963), 734-752. MR 28 #4102. MR 0160893 (28:4102)
- [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]
- -, Groups of homeomorphisms of a simply connected space, Ann. of Math. (2) 79 (1964), 473-488. MR 28 #4058. MR 0160848 (28:4058)
- [10]
- -, The classification of non-euclidean plane crystallographic groups, Canad. J. Math. 19 (1967), 1192-1205. MR 36 #3890. MR 0220838 (36:3890)
- [11]
- J. Nielsen, Untersuchungen zur Topologie der geschlossenen zweiseitigen Flachen, Acta Math. 50 (1927), 264-275. MR 1555256
- [12]
- H. E. Rauch, A transcendental view of the space of algebraic Riemann surfaces, Bull. Amer. Math. Soc. 71 (1965), 1-39. MR 35 #4403. MR 0213543 (35:4403)
- [13]
- A. Weil, On discrete subgroups of Lie groups, Ann. of Math. (2) 72 (1960), 369-384. MR 25 #1241. MR 0137792 (25:1241)
- [14]
- H. Zieschang, Über Automorphismen ebener diskontinuierlicher Gruppen, Math. Ann. 166 (1966), 148-167. MR 34 #1403. MR 0201521 (34:1403)
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Additional Information
DOI:
http://dx.doi.org/10.1090/S0002-9947-1971-0297994-0
PII:
S 0002-9947(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
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