Inequivalent topologies on the Teichmüller space of the flute surface

Evren, Özgür

doi:10.1090/proc/13412

Inequivalent topologies on the Teichmüller space of the flute surface
HTML articles powered by AMS MathViewer

by Özgür Evren PDF

Proc. Amer. Math. Soc. 145 (2017), 2607-2621 Request permission

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

A. Basmajian and D. Saric, Geodesically Complete Hyperbolic Structures, ArXiv Mathematics e-prints arXiv:math.GT/1508.02280 (2015).
Ara Basmajian, Hyperbolic structures for surfaces of infinite type, Trans. Amer. Math. Soc. 336 (1993), no. 1, 421–444. MR 1087051, DOI 10.1090/S0002-9947-1993-1087051-2
Ara Basmajian, Large parameter spaces of quasiconformally distinct hyperbolic structures, J. Anal. Math. 71 (1997), 75–85. MR 1454244, DOI 10.1007/BF02788023
Ara Basmajian and Youngju Kim, Geometrically infinite surfaces with discrete length spectra, Geom. Dedicata 137 (2008), 219–240. MR 2449153, DOI 10.1007/s10711-008-9294-5
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, DOI 10.1112/jlms/jdm052
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
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, DOI 10.1007/978-4-431-68174-8
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, DOI 10.2996/kmj/1309829545
Zhong Li, Teichmüller metric and length spectrums of Riemann surfaces, Sci. Sinica Ser. A 29 (1986), no. 3, 265–274. MR 855233
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
Liu Lixin, On the length spectrum of non-compact Riemann surfaces, Ann. Acad. Sci. Fenn. Math. 24 (1999), no. 1, 11–22. MR 1678001
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, DOI 10.2996/kmj/1073670609
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
Tuomas Sorvali, The boundary mapping induced by an isomorphism of covering groups, Ann. Acad. Sci. Fenn. Ser. A. I. 526 (1972), 31. MR 328066
Tuomas Sorvali, On Teichmüller spaces of tori, Ann. Acad. Sci. Fenn. Ser. A I Math. 1 (1975), no. 1, 7–11. MR 0435389
W. P. Thurston, Minimal stretch maps between hyperbolic surfaces, ArXiv Mathematics e-prints arXiv:math.GT/9801039 (1986).
Scott Wolpert, The length spectra as moduli for compact Riemann surfaces, Ann. of Math. (2) 109 (1979), no. 2, 323–351. MR 528966, DOI 10.2307/1971114