## On the length spectrum Teichmüller spaces of Riemann surfaces of infinite type

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- by Erina Kinjo
- Conform. Geom. Dyn.
**22**(2018), 1-14 - DOI: https://doi.org/10.1090/ecgd/316
- Published electronically: February 26, 2018
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## Abstract:

On the Teichmüller space $T(R_0)$ of a hyperbolic Riemann surface $R_0$, we consider the length spectrum metric $d_L$, which measures the difference of hyperbolic structures of Riemann surfaces. It is known that if $R_0$ is of finite type, then $d_L$ defines the same topology as that of Teichmüller metric $d_T$ on $T(R_0)$. In 2003, H. Shiga extended the discussion to the Teichmüller spaces of Riemann surfaces of infinite type and proved that the two metrics define the same topology on $T(R_0)$ if $R_0$ satisfies some geometric condition. After that, Alessandrini-Liu-Papadopoulos-Su proved that for the Riemann surface satisfying Shiga’s condition, the identity map between the two metric spaces is locally bi-Lipschitz.

In this paper, we extend their results; that is, we show that if $R_0$ has bounded geometry, then the identity map $(T(R_0),d_L) \to (T(R_0),d_T)$ is locally bi-Lipschitz.

## References

- 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 - D. Alessandrini, L. Liu, A. Papadopoulos, and W. Su,
*On local comparison between various metrics on Teichmüller spaces*, Geom. Dedicata**157**(2012), 91–110. MR**2893480**, DOI 10.1007/s10711-011-9601-4 - 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** - Adam Lawrence Epstein,
*Effectiveness of Teichmüller modular groups*, In the tradition of Ahlfors and Bers (Stony Brook, NY, 1998) Contemp. Math., vol. 256, Amer. Math. Soc., Providence, RI, 2000, pp. 69–74. MR**1759670**, DOI 10.1090/conm/256/03997 - 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** - Benson Farb and Dan Margalit,
*A primer on mapping class groups*, Princeton Mathematical Series, vol. 49, Princeton University Press, Princeton, NJ, 2012. MR**2850125** - 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 - Erina Kinjo,
*On the length spectrum metric in infinite dimensional Teichmüller spaces*, Ann. Acad. Sci. Fenn. Math.**39**(2014), no. 1, 349–360. MR**3186819**, DOI 10.5186/aasfm.2014.3925 - Zhong Li,
*Teichmüller metric and length spectrums of Riemann surfaces*, Sci. Sinica Ser. A**29**(1986), no. 3, 265–274. MR**855233** - Liu Lixin,
*On the length spectrum of non-compact Riemann surfaces*, Ann. Acad. Sci. Fenn. Math.**24**(1999), no. 1, 11–22. MR**1678001** - Lixin Liu and Athanase Papadopoulos,
*Some metrics on Teichmüller spaces of surfaces of infinite type*, Trans. Amer. Math. Soc.**363**(2011), no. 8, 4109–4134. MR**2792982**, DOI 10.1090/S0002-9947-2011-05090-7 - 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** - 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** - W. P. Thurston,
*Minimal stretch maps between hyperbolic surfaces*; 1986 preprint converted to 1998 eprint, http://arxiv.org/pdf/math/9801039

## Bibliographic Information

**Erina Kinjo**- Affiliation: Department of Mathematics, Tokyo Institute of Technology, Oh-okayama 2-12-1, Meguro-ku, Tokyo 152-8551, Japan
- MR Author ID: 942104
- Email: kinjo.e.aa@m.titech.ac.jp
- Received by editor(s): September 23, 2016
- Received by editor(s) in revised form: October 2, 2017
- Published electronically: February 26, 2018
- © Copyright 2018 American Mathematical Society
- Journal: Conform. Geom. Dyn.
**22**(2018), 1-14 - MSC (2010): Primary 30F60; Secondary 32G15
- DOI: https://doi.org/10.1090/ecgd/316
- MathSciNet review: 3768110