Pseudo-periodic homeomorphisms and degeneration of Riemann surfaces. (English) Zbl 0797.30036
In this paper two theorems are announced. The first theorem gives a classification of all topological types of degenerate fibres, which can appear in a minimal degenerating family of Riemann surfaces, as the subset of the set of conjugacy classes in the mapping class group of a closed surface of genus \(g \geq 2\); every conjugacy class of this subset has as represent a pseudoperiodic homeomorphism of negative twist. The second theorem describes a complete set of conjugacy invariants for the mapping classes of pseudoperiodic homeomorphisms of negative twist.
Reviewer: A.Duma (Hagen)
MSC:
| 30F99 | Riemann surfaces |
| 32S50 | Topological aspects of complex singularities: Lefschetz theorems, topological classification, invariants |
References:
| [1] | Lipman Bers, Spaces of degenerating Riemann surfaces, Discontinuous groups and Riemann surfaces (Proc. Conf., Univ. Maryland, College Park, Md., 1973) Princeton Univ. Press, Princeton, N.J., 1974, pp. 43 – 55. Ann. of Math. Studies, No. 79. · Zbl 0045.42501 |
| [2] | Clifford J. Earle and Patricia L. Sipe, Families of Riemann surfaces over the punctured disk, Pacific J. Math. 150 (1991), no. 1, 79 – 96. · Zbl 0734.30039 |
| [3] | Jane Gilman, On the Nielsen type and the classification for the mapping class group, Adv. in Math. 40 (1981), no. 1, 68 – 96. · Zbl 0474.57005 · doi:10.1016/0001-8708(81)90033-5 |
| [4] | Yôichi Imayoshi, Holomorphic families of Riemann surfaces and Teichmüller spaces, Riemann surfaces and related topics: Proceedings of the 1978 Stony Brook Conference (State Univ. New York, Stony Brook, N.Y., 1978) Ann. of Math. Stud., vol. 97, Princeton Univ. Press, Princeton, N.J., 1981, pp. 277 – 300. Yôichi Imayoshi, Holomorphic families of Riemann surfaces and Teichmüller spaces. II, Tôhoku Math. J. (2) 31 (1979), no. 4, 469 – 489. , https://doi.org/10.2748/tmj/1178229731 Yôichi Imayoshi, Holomorphic families of Riemann surfaces and Teichmüller spaces. III. Bimeromorphic embedding of algebraic surfaces into projective spaces by automorphic forms, Tôhoku Math. J. (2) 33 (1981), no. 2, 227 – 247. · Zbl 0504.32019 · doi:10.2748/tmj/1178229451 |
| [5] | K. Kodaira, On compact analytic surfaces. II, Ann. of Math. (2) 77 (1963), 563-626. · Zbl 0118.15802 |
| [6] | Y. Matsumoto and J. M. Montesinos-Amilibia, Singular fibers and pseudo-periodic surface automorphisms, Conference Report for Knots 90, Osaka, 1990, pp. 50-51. |
| [7] | -, Pseudo-periodic maps and degeneration of Riemann surfaces. I, II, preprint, Univ. of Tokyo and Univ. Complutense de Madrid, 1991/1992. |
| [8] | Yukihiko Namikawa and Kenji Ueno, The complete classification of fibres in pencils of curves of genus two, Manuscripta Math. 9 (1973), 143 – 186. · Zbl 0263.14007 · doi:10.1007/BF01297652 |
| [9] | J. Nielsen, Die structur periodischer transformationen von Flächen, Mat.-Fys. Medd. Danske Vid. Selsk 15 (1937); English transl. by J. Stillwell, The structure of periodic surface transformations, Collected Papers 2, Birkhäuser, 1986. |
| [10] | Jakob Nielsen, Surface transformation classes of algebraically finite type, Danske Vid. Selsk. Math.-Phys. Medd. 21 (1944), no. 2, 89. · Zbl 0063.05952 |
| [11] | Takayuki Oda, A note on ramification of the Galois representation on the fundamental group of an algebraic curve. II, J. Number Theory 53 (1995), no. 2, 342 – 355. · Zbl 0844.14013 · doi:10.1006/jnth.1995.1095 |
| [12] | Hiroshige Shiga and Harumi Tanigawa, On the Maskit coordinates of Teichmüller spaces and modular transformations, Kodai Math. J. 12 (1989), no. 3, 437 – 443. · Zbl 0698.32013 · doi:10.2996/kmj/1138039108 |
| [13] | Itiro Tamura, Foliations and spinnable structures on manifolds, Ann. Inst. Fourier (Grenoble) 23 (1973), no. 2, 197 – 214 (English, with French summary). Colloque International sur l’Analyse et la Topologie Différentielle (Colloques Internationaux du Centre National de la Recherche Scientifique, Strasbourg, 1972). · Zbl 0269.57012 |
| [14] | William P. Thurston, On the geometry and dynamics of diffeomorphisms of surfaces, Bull. Amer. Math. Soc. (N.S.) 19 (1988), no. 2, 417 – 431. · Zbl 0674.57008 |
| [15] | H. E. Winkelnkemper, Manifolds as open books, Bull. Amer. Math. Soc. 79 (1973), 45 – 51. · Zbl 0269.57011 |
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