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

HTML articles powered by AMS MathViewer

- by Ege Fujikawa and Masahiko Taniguchi PDF
- Conform. Geom. Dyn.
**21**(2017), 64-77 Request permission

## 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

- 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**, DOI 10.1007/BF02392590 - 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**, DOI 10.24033/asens.1491 - 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**, DOI 10.1007/978-1-4613-9602-4_{1}2 - 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**, DOI 10.1090/conm/355/06455 - Ege Fujikawa,
*Pure mapping class group acting on Teichmüller space*, Conform. Geom. Dyn.**12**(2008), 227–239. MR**2466018**, DOI 10.1090/S1088-4173-08-00188-4 - 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**, DOI 10.2969/jmsj/1190905449 - Irwin Kra,
*On spaces of Kleinian groups*, Comment. Math. Helv.**47**(1972), 53–69. MR**306485**, DOI 10.1007/BF02566788 - Gregory Stephen Lieb,
*Holomorphic motions and Teichmuller space*, ProQuest LLC, Ann Arbor, MI, 1989. Thesis (Ph.D.)–Cornell University. MR**2638376** - Albert Marden,
*The geometry of finitely generated kleinian groups*, Ann. of Math. (2)**99**(1974), 383–462. MR**349992**, DOI 10.2307/1971059 - Bernard Maskit,
*Isomorphisms of function groups*, J. Analyse Math.**32**(1977), 63–82. MR**463430**, DOI 10.1007/BF02803575 - 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** - 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**, DOI 10.4171/117-1/16 - Sudeb Mitra,
*Teichmüller contraction in the Teichmüller space of a closed set in the sphere*, Israel J. Math.**125**(2001), 45–51. MR**1853804**, DOI 10.1007/BF02773373 - 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**, DOI 10.1007/BF02392543 - M. Taniguchi,
*Teichmüller space of a countable set of points on the Riemann sphere*, Filomat, to appear.

## Additional Information

**Ege Fujikawa**- Affiliation: Department of Mathematics, Chiba University, Inage-ku, Chiba 263-8522, Japan
- MR Author ID: 706593
- Email: fujikawa@math.s.chiba-u.ac.jp
**Masahiko Taniguchi**- Affiliation: Department of Mathematics, Nara Women’s University, Nara 630-8506, Japan
- MR Author ID: 192108
- Email: tanig@cc.nara-wu.ac.jp
- 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 - © Copyright 2017 American Mathematical Society
- Journal: Conform. Geom. Dyn.
**21**(2017), 64-77 - MSC (2010): Primary 30F60; Secondary 32G15
- DOI: https://doi.org/10.1090/ecgd/301
- MathSciNet review: 3603961