Earthquakes in the length-spectrum Teichmüller spaces
HTML articles powered by AMS MathViewer
- by Dragomir Šarić PDF
- Proc. Amer. Math. Soc. 143 (2015), 1531-1543 Request permission
Abstract:
Let $X_0$ be a complete hyperbolic surface of infinite type that has a geodesic pants decomposition with cuff lengths bounded above. The length spectrum Teichmüller space $T_{ls}(X_0)$ consists of homotopy classes of hyperbolic metrics on $X_0$ such that the ratios of the corresponding simple closed geodesic for the hyperbolic metric on $X_0$ and for the other hyperbolic metric are bounded from below away from $0$ and from above away from $\infty$. This paper studies earthquakes in the length spectrum Teichmüller space $T_{ls}(X_0)$. We find a necessary condition and several sufficient conditions on the earthquake measure $\mu$ such that the corresponding earthquake $E^{\mu }$ describes a hyperbolic metric on $X_0$ which is in the length spectrum Teichmüller space. Moreover, we give examples of earthquake paths $t\mapsto E^{t\mu }$, for $t\geq 0$, such that $E^{t\mu }\in T_{ls}(X_0)$ for $0\leq t<t_0$, $E^{t_0\mu }\notin T_{ls}(X_0)$ and $E^{t\mu }\in T_{ls}(X_0)$ for $t>t_0$.References
- D. Alessandrini, L. Liu, A. Pappadopoulos, and W. Su, On the inclusion of the quasiconformal Teichmüller space into the length-spectrum Teichmüller space, preprint, available on arXiv.
- Daniele Alessandrini, Lixin Liu, Athanase Papadopoulos, Weixu Su, and Zongliang Sun, On Fenchel-Nielsen coordinates on Teichmüller spaces of surfaces of infinite type, Ann. Acad. Sci. Fenn. Math. 36 (2011), no. 2, 621–659. MR 2865518, DOI 10.5186/aasfm.2011.3637
- Christopher J. Bishop, Quasiconformal mappings of $Y$-pieces, Rev. Mat. Iberoamericana 18 (2002), no. 3, 627–652. MR 1954866, DOI 10.4171/RMI/330
- Peter Buser, Geometry and spectra of compact Riemann surfaces, Progress in Mathematics, vol. 106, Birkhäuser Boston, Inc., Boston, MA, 1992. MR 1183224
- 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
- D. B. A. Epstein and A. Marden, Convex hulls in hyperbolic space, a theorem of Sullivan, and measured pleated surfaces, Analytical and geometric aspects of hyperbolic space (Coventry/Durham, 1984) London Math. Soc. Lecture Note Ser., vol. 111, Cambridge Univ. Press, Cambridge, 1987, pp. 113–253. MR 903852
- D. B. A. Epstein, A. Marden, and V. Markovic, Quasiconformal homeomorphisms and the convex hull boundary, Ann. of Math. (2) 159 (2004), no. 1, 305–336. MR 2052356, DOI 10.4007/annals.2004.159.305
- F. P. Gardiner, J. Hu, and N. Lakic, Earthquake curves, Complex manifolds and hyperbolic geometry (Guanajuato, 2001) Contemp. Math., vol. 311, Amer. Math. Soc., Providence, RI, 2002, pp. 141–195. MR 1940169, DOI 10.1090/conm/311/05452
- Jun Hu, Earthquake measure and cross-ratio distortion, In the tradition of Ahlfors and Bers, III, Contemp. Math., vol. 355, Amer. Math. Soc., Providence, RI, 2004, pp. 285–308. MR 2145070, DOI 10.1090/conm/355/06459
- Steven P. Kerckhoff, The Nielsen realization problem, Ann. of Math. (2) 117 (1983), no. 2, 235–265. MR 690845, DOI 10.2307/2007076
- Yair N. Minsky, Extremal length estimates and product regions in Teichmüller space, Duke Math. J. 83 (1996), no. 2, 249–286. MR 1390649, DOI 10.1215/S0012-7094-96-08310-6
- Hideki Miyachi and Dragomir Šarić, Uniform weak$^*$ topology and earthquakes in the hyperbolic plane, Proc. Lond. Math. Soc. (3) 105 (2012), no. 6, 1123–1148. MR 3004100, DOI 10.1112/plms/pds026
- Dragomir Šarić, Real and complex earthquakes, Trans. Amer. Math. Soc. 358 (2006), no. 1, 233–249. MR 2171231, DOI 10.1090/S0002-9947-05-03651-2
- Dragomir Šarić, Bounded earthquakes, Proc. Amer. Math. Soc. 136 (2008), no. 3, 889–897. MR 2361861, DOI 10.1090/S0002-9939-07-09146-0
- D. Šarić, Fenchel-Nielsen coordinates on upper bounded pants decompositions, preprint available on arXiv, accepted to Math. Proc. Camb. Phil. Soc.
- 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
- William P. Thurston, Earthquakes in two-dimensional hyperbolic geometry, Low-dimensional topology and Kleinian groups (Coventry/Durham, 1984) London Math. Soc. Lecture Note Ser., vol. 112, Cambridge Univ. Press, Cambridge, 1986, pp. 91–112. MR 903860
Additional Information
- Dragomir Šarić
- Affiliation: Department of Mathematics, Queens College of CUNY, 65-30 Kissena Boulevard, Flushing, New York 11367; and Mathematics Ph.D. Program, The CUNY Graduate Center, 365 Fifth Avenue, New York, New York 10016-4309
- Email: Dragomir.Saric@qc.cuny.edu
- Received by editor(s): December 1, 2012
- Received by editor(s) in revised form: April 24, 2013
- Published electronically: December 4, 2014
- Additional Notes: This research was partially supported by National Science Foundation grant DMS 1102440.
- Communicated by: Michael Wolf
- © Copyright 2014
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Proc. Amer. Math. Soc. 143 (2015), 1531-1543
- MSC (2010): Primary 30F60; Secondary 32G15
- DOI: https://doi.org/10.1090/S0002-9939-2014-12242-8
- MathSciNet review: 3314067