Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 
 

 

Inequivalent topologies on the Teichmüller space of the flute surface


Author: Özgür Evren
Journal: Proc. Amer. Math. Soc. 145 (2017), 2607-2621
MSC (2010): Primary 30F60
DOI: https://doi.org/10.1090/proc/13412
Published electronically: December 27, 2016
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: The topology defined by the symmetrized Lipschitz metric on the Teichmüller space of an infinite type surface, in contrast to finite type surfaces, need not be the same as the topology defined by the Teichmüller metric. In this paper, we study the equivalence of these topologies on a particular kind of infinite type surface, called the flute surface.

Following a construction by Shiga and using additional hyperbolic geometric estimates, we obtain sufficient conditions in terms of length parameters for these two metrics to be topologically inequivalent. Next, we construct infinite parameter families of quasiconformally distinct flute surfaces with the property that the symmetrized Lipschitz metric is not topologically equivalent to the Teichmüller metric.


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

  • [1] A. Basmajian and D. Saric, Geodesically Complete Hyperbolic Structures, ArXiv Mathematics e-prints arXiv:math.GT/1508.02280 (2015).
  • [2] Ara Basmajian, Hyperbolic structures for surfaces of infinite type, Trans. Amer. Math. Soc. 336 (1993), no. 1, 421-444. MR 1087051, https://doi.org/10.2307/2154353
  • [3] Ara Basmajian, Large parameter spaces of quasiconformally distinct hyperbolic structures, J. Anal. Math. 71 (1997), 75-85. MR 1454244, https://doi.org/10.1007/BF02788023
  • [4] Ara Basmajian and Youngju Kim, Geometrically infinite surfaces with discrete length spectra, Geom. Dedicata 137 (2008), 219-240. MR 2449153, https://doi.org/10.1007/s10711-008-9294-5
  • [5] Young-Eun Choi and Kasra Rafi, Comparison between Teichmüller and Lipschitz metrics, J. Lond. Math. Soc. (2) 76 (2007), no. 3, 739-756. MR 2377122, https://doi.org/10.1112/jlms/jdm052
  • [6] Ozgur Evren, The Length Spectrum Metric on the Teichmuller Space of a Flute Surface, ProQuest LLC, Ann Arbor, MI, 2013. Thesis (Ph.D.)-City University of New York. MR 3152661
  • [7] Y. Imayoshi and M. Taniguchi, An introduction to Teichmüller spaces, Springer-Verlag, Tokyo, 1992. Translated and revised from the Japanese by the authors. MR 1215481
  • [8] Erina Kinjo, On Teichmüller metric and the length spectrums of topologically infinite Riemann surfaces, Kodai Math. J. 34 (2011), no. 2, 179-190. MR 2811639, https://doi.org/10.2996/kmj/1309829545
  • [9] Zhong Li, Teichmüller metric and length spectrums of Riemann surfaces, Sci. Sinica Ser. A 29 (1986), no. 3, 265-274. MR 855233
  • [10] Lixin Liu, Zongliang Sun, and Hanbai Wei, Topological equivalence of metrics in Teichmüller space, Ann. Acad. Sci. Fenn. Math. 33 (2008), no. 1, 159-170. MR 2386845
  • [11] Liu Lixin, On the length spectrum of non-compact Riemann surfaces, Ann. Acad. Sci. Fenn. Math. 24 (1999), no. 1, 11-22. MR 1678001
  • [12] Katsuhiko Matsuzaki, The infinite direct product of Dehn twists acting on infinite dimensional Teichmüller spaces, Kodai Math. J. 26 (2003), no. 3, 279-287. MR 2018722, https://doi.org/10.2996/kmj/1073670609
  • [13] Hiroshige Shiga, On a distance defined by the length spectrum of Teichmüller space, Ann. Acad. Sci. Fenn. Math. 28 (2003), no. 2, 315-326. MR 1996441
  • [14] Tuomas Sorvali, The boundary mapping induced by an isomorphism of covering groups, Ann. Acad. Sci. Fenn. Ser. A I 526 (1972), 31. MR 0328066
  • [15] Tuomas Sorvali, On Teichmüller spaces of tori, Ann. Acad. Sci. Fenn. Ser. A I Math. 1 (1975), no. 1, 7-11. MR 0435389
  • [16] W. P. Thurston, Minimal stretch maps between hyperbolic surfaces, ArXiv Mathematics e-prints arXiv:math.GT/9801039 (1986).
  • [17] Scott Wolpert, The length spectra as moduli for compact Riemann surfaces, Ann. of Math. (2) 109 (1979), no. 2, 323-351. MR 528966, https://doi.org/10.2307/1971114

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2010): 30F60

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


Additional Information

Özgür Evren
Affiliation: Department of Mathematics, Galatasaray University, Istanbul, Turkey
Email: oevren@gradcenter.cuny.edu

DOI: https://doi.org/10.1090/proc/13412
Keywords: The symmetrized Lipschitz metric, infinite type Riemann surfaces
Received by editor(s): September 25, 2015
Received by editor(s) in revised form: July 29, 2016
Published electronically: December 27, 2016
Additional Notes: The author was supported by Tübitak through 1001 - Scientific and Technological Research Projects Funding Program, project 114R073.
Communicated by: Ken Ono
Article copyright: © Copyright 2016 American Mathematical Society

American Mathematical Society