Remote Access Conformal Geometry and Dynamics
Green Open Access

Conformal Geometry and Dynamics

ISSN 1088-4173

 
 

 

The Teichmüller space of a countable set of points on a Riemann surface


Authors: Ege Fujikawa and Masahiko Taniguchi
Journal: Conform. Geom. Dyn. 21 (2017), 64-77
MSC (2010): Primary 30F60; Secondary 32G15
DOI: https://doi.org/10.1090/ecgd/301
Published electronically: February 1, 2017
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: We introduce the quasiconformal deformation space of an ordered countable set of an infinite number of points on a Riemann surface and give certain conditions under which it admits a complex structure via Teichmüller spaces of associated subsurfaces with the complement of the set of points. In a similar fashion, we give another definition of the quasiconformal deformation space of a finitely generated Kleinian group.


References [Enhancements On Off] (What's this?)

  • [1] Adrien Douady and Clifford J. Earle, Conformally natural extension of homeomorphisms of the circle, Acta Math. 157 (1986), no. 1-2, 23-48. MR 857678, https://doi.org/10.1007/BF02392590
  • [2] Adrien Douady and John Hamal Hubbard, On the dynamics of polynomial-like mappings, Ann. Sci. École Norm. Sup. (4) 18 (1985), no. 2, 287-343. MR 816367
  • [3] Clifford J. Earle and Curt McMullen, Quasiconformal isotopies, Holomorphic functions and moduli, Vol. I (Berkeley, CA, 1986) Math. Sci. Res. Inst. Publ., vol. 10, Springer, New York, 1988, pp. 143-154. MR 955816, https://doi.org/10.1007/978-1-4613-9602-4_12
  • [4] Ege Fujikawa, Modular groups acting on infinite dimensional Teichmüller spaces, In the tradition of Ahlfors and Bers, III, Contemp. Math., vol. 355, Amer. Math. Soc., Providence, RI, 2004, pp. 239-253. MR 2145066, https://doi.org/10.1090/conm/355/06455
  • [5] Ege Fujikawa, Pure mapping class group acting on Teichmüller space, Conform. Geom. Dyn. 12 (2008), 227-239. MR 2466018, https://doi.org/10.1090/S1088-4173-08-00188-4
  • [6] Ege Fujikawa, Hiroshige Shiga, and Masahiko Taniguchi, On the action of the mapping class group for Riemann surfaces of infinite type, J. Math. Soc. Japan 56 (2004), no. 4, 1069-1086. MR 2091417, https://doi.org/10.2969/jmsj/1190905449
  • [7] Irwin Kra, On spaces of Kleinian groups, Comment. Math. Helv. 47 (1972), 53-69. MR 0306485, https://doi.org/10.1007/BF02566788
  • [8] Gregory Stephen Lieb, Holomorphic motions and Teichmuller space, ProQuest LLC, Ann Arbor, MI, 1989. Thesis (Ph.D.)-Cornell University. MR 2638376
  • [9] Albert Marden, The geometry of finitely generated Kleinian groups, Ann. of Math. (2) 99 (1974), 383-462. MR 0349992, https://doi.org/10.2307/1971059
  • [10] Bernard Maskit, Isomorphisms of function groups, J. Analyse Math. 32 (1977), 63-82. MR 0463430, https://doi.org/10.1007/BF02803575
  • [11] Katsuhiko Matsuzaki and Masahiko Taniguchi, Hyperbolic manifolds and Kleinian groups, Oxford Mathematical Monographs, The Clarendon Press, Oxford University Press, New York, 1998. Oxford Science Publications. MR 1638795
  • [12] Katsuhiko Matsuzaki, Infinite-dimensional Teichmüller spaces and modular groups, Handbook of Teichmüller theory. Vol. IV, IRMA Lect. Math. Theor. Phys., vol. 19, Eur. Math. Soc., Zürich, 2014, pp. 681-716. MR 3289713, https://doi.org/10.4171/117-1/16
  • [13] Sudeb Mitra, Teichmüller contraction in the Teichmüller space of a closed set in the sphere, Israel J. Math. 125 (2001), 45-51. MR 1853804, https://doi.org/10.1007/BF02773373
  • [14] Dennis Sullivan, Quasiconformal homeomorphisms and dynamics. II. Structural stability implies hyperbolicity for Kleinian groups, Acta Math. 155 (1985), no. 3-4, 243-260. MR 806415, https://doi.org/10.1007/BF02392543
  • [15] M. Taniguchi,
    Teichmüller space of a countable set of points on the Riemann sphere,
    Filomat, to appear.

Similar Articles

Retrieve articles in Conformal Geometry and Dynamics of the American Mathematical Society with MSC (2010): 30F60, 32G15

Retrieve articles in all journals with MSC (2010): 30F60, 32G15


Additional Information

Ege Fujikawa
Affiliation: Department of Mathematics, Chiba University, Inage-ku, Chiba 263-8522, Japan
Email: fujikawa@math.s.chiba-u.ac.jp

Masahiko Taniguchi
Affiliation: Department of Mathematics, Nara Women’s University, Nara 630-8506, Japan
Email: tanig@cc.nara-wu.ac.jp

DOI: https://doi.org/10.1090/ecgd/301
Received by editor(s): August 18, 2016
Received by editor(s) in revised form: January 13, 2017
Published electronically: February 1, 2017
Additional Notes: The first author was partially supported by Grants-in-Aid for Scientific Research (C) Grant No. 25400127
The second author was partially supported by Grants-in-Aid for Scientific Research (C) Grant No. 15K04925
Article copyright: © Copyright 2017 American Mathematical Society

American Mathematical Society