Hostname: page-component-586b7cd67f-tf8b9 Total loading time: 0 Render date: 2024-11-26T07:42:55.986Z Has data issue: false hasContentIssue false
  • English
  • Français

Geodesically Complete Hyperbolic Structures

Published online by Cambridge University Press:  20 November 2017

ARA BASMAJIAN  and
DRAGOMIR ŠARIĆ
ARA BASMAJIAN
Affiliation:
Ph.D. Program in Mathematics, The Graduate Center of the City University of New York, NY 10016, USA. Department of Mathematics and Statistics, Hunter College, 695 Park Ave., New York, NY 10065, USA. e-mail: [email protected]
DRAGOMIR ŠARIĆ
Affiliation:
Ph.D. Program in Mathematics, The Graduate Center of the City University of New York, NY 10016, USA. Department of Mathematics, Queens College, 65-30 Kissena Blvd., Flushing, NY 11367, USA. e-mail: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

In the first part of this work we explore the geometry of infinite type surfaces and the relationship between its convex core and space of ends. In particular, we give a geometric proof of a Theorem due to Alvarez and Rodriguez that a geodesically complete hyperbolic surface is made up of its convex core with funnels attached along the simple closed geodesic components and half-planes attached along simple open geodesic components. We next consider gluing infinitely many pairs of pants along their cuffs to obtain an infinite hyperbolic surface. We prove that there always exists a choice of twists in the gluings such that the surface is complete regardless of the size of the cuffs. This generalises the examples of Matsuzaki.

In the second part we consider complete hyperbolic flute surfaces with rapidly increasing cuff lengths and prove that the corresponding quasiconformal Teichmüller space is incomplete in the length spectrum metric. Moreover, we describe the twist coordinates and convergence in terms of the twist coordinates on the closure of the quasiconformal Teichmüller space.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 166 , Issue 2 , March 2019 , pp. 219 - 242
Copyright
Copyright © Cambridge Philosophical Society 2017 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Footnotes

†

Supported in part by a PSC-CUNY Grant and a Simons foundation grant.

‡

Supported by the National Science Foundation grant DMS 1102440, PSC-CUNY grant, and a Simons foundation grant.

References

REFERENCES

[1] Ahlfors, L. V. Complex Analysis (McGraw Hill, 3rd ed., 1979).Google Scholar
[2] Alessandrini, D., Liu, L., Papadopoulos, A. and Su, W.. On the inclusion of the quasiconformal Teichmüller space into the length-spectrum Teichmüller space. Monatsh. Math. 179 (2016), no. 2, 165–189.Google Scholar
[3] Alessandrini, D., Liu, L., Papadopoulos, A. and Su, W.. On various Teichmüller spaces of a surface of infinite topological type. Proc. Amer. Math. Soc. 140 (2012), 561–574.Google Scholar
[4] Alessandrini, D., Liu, L., Papadopoulos, A., Su, W. and Sun, Z.. On Fenchel–Nielsen coordinates on Teichmüller spaces of surfaces of infinite type. Ann. Acad. Sci. Fenn. Math. 36 (2011), no. 2, 621–659.Google Scholar
[5] Alvarez, V. and Rodriguez, J. M.. Structure Theorems for Riemann and topological surfaces. J. London Math. Soc. (2) 69 (2004), 153–168.Google Scholar
[6] Basmajian, A. Hyperbolic structures for surfaces of infinite type. Trans. Amer. Math. Soc. 336, no. 1 (March 1993), 421–444.Google Scholar
[7] Basmajian, A. Large parameter spaces of quasiconformally distinct hyperbolic structures. J. Anal. Math. 71 (1997), 75–85.Google Scholar
[8] Basmajian, A. and Kim, Y.. Geometrically infinite surfaces with discrete length spectra. Geom. Dedicata 137 (2008), 219–240.Google Scholar
[9] Beardon, A. The geometry of discrete groups. Graduate Texts in Mathematics, 91 (Springer–Verlag, New York, 1983).Google Scholar
[10] Buser, P. Geometry and spectra of compact Riemann surfaces. Progr. Math. 106 (Birkhuser Boston, Inc., Boston, MA, 1992), xiv+454 pp. ISBN: 0-8176-3406-1Google Scholar
[11] Epstein, D. B. A., Marden, A. and Markovic, V.. Quasiconformal homeomorphisms and the convex hull boundary. Ann. of Math. (2) 159 (2004), no. 1, 305–336.Google Scholar
[12] Haas, A. and Susskind, P.. The geometry at infinity of a hyperbolic Reimann surface of infinite type. Geom. Dedicata 130, Issue 1 (2007), pp 1–24.Google Scholar
[13] Kérékjárto, B. Vorlesungen über Topologie I (Springer, Berlin, 1923).Google Scholar
[14] Kinjo, E. On Teichmüller metric and the length spectrums of topologically infinite Riemann surfaces. Kodai Math. J. 34 (2011), no. 2, 179–190.Google Scholar
[15] Matsuzaki, K. A countable Teichmüller modular group. Trans. Amer. Math. Soc. 357, 8 (2004), 3119–3131.Google Scholar
[16] Richards, I. On the classification of noncompact surfaces. Trans. Amer. Math. Soc. 106 (1963) 259–269.Google Scholar
[17] Šarić, D.. Real and complex earthquakes. Trans. Amer. Math. Soc. 358 (2006), no. 1, 233–249.Google Scholar
[18] Šarić, D.. Fenchel–Nielsen coordinates on upper bounded pants decompositions. To appear Math. Proc. Camb. Phil. Soc., arXiv:1209.5819.Google Scholar
[19] Shiga, H. On a distance defined by the length spectrum of Teichmüller space. Ann. Acad. Sci. Fenn. Math. 28 (2003), no. 2, 315–326.Google Scholar
[20] Sorvali, T. On the dilatation of isomorphisms between covering groups. Ann. Acad. Sci. Fenn. Ser. A, I 551 (1973), 1–15.Google Scholar
[21] Teichmüller, O. Extremale quasikonforme Abbildungen und quadratische Differentiale. Abh. Preuss. Akad. Wiss. Math.-Nat. Kl. 1939 (1940), no. 22, 197pp.Google Scholar
[22] Wolpert, S. The length spectra as moduli for compact Riemann surfaces. Ann. of Math. (2) 109 (1979), no. 2, 323–351.Google Scholar