Almost-isometry between the Teichmüller metric and the length-spectrum metric on reduced moduli space for surfaces with boundary

Liu, L.; Shiga, H.; Su, W.; Zhong, Y.

doi:10.1090/tran/6877

Almost-isometry between the Teichmüller metric and the length-spectrum metric on reduced moduli space for surfaces with boundary
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by L. Liu, H. Shiga, W. Su and Y. Zhong PDF

Trans. Amer. Math. Soc. 369 (2017), 6429-6464 Request permission

Abstract:

We show that the Teichmüller metric and the length-spectrum metric are almost-isometric on moduli space of hyperbolic surfaces with geodesic boundaries whose lengths are bounded above.

References

D. Alessandrini, L. Liu, A. Papadopoulos, and W. Su, The horofunction compactification of Teichmüller spaces of surfaces with boundary, Topology Appl. 208 (2016), 160–191. MR 3506976, DOI 10.1016/j.topol.2016.05.011
Lipman Bers, An inequality for Riemann surfaces, Differential geometry and complex analysis, Springer, Berlin, 1985, pp. 87–93. MR 780038
Francesco Bonsante, Kirill Krasnov, and Jean-Marc Schlenker, Multi-black holes and earthquakes on Riemann surfaces with boundaries, Int. Math. Res. Not. IMRN 3 (2011), 487–552. MR 2764871, DOI 10.1093/imrn/rnq070
Martin Bridgeman, Orthospectra of geodesic laminations and dilogarithm identities on moduli space, Geom. Topol. 15 (2011), no. 2, 707–733. MR 2800364, DOI 10.2140/gt.2011.15.707
Peter Buser, Geometry and spectra of compact Riemann surfaces, Progress in Mathematics, vol. 106, Birkhäuser Boston, Inc., Boston, MA, 1992. MR 1183224
Jeffrey Danciger, François Guéritaud, and Fanny Kassel, Margulis spacetimes via the arc complex, Invent. Math. 204 (2016), no. 1, 133–193. MR 3480555, DOI 10.1007/s00222-015-0610-z
Benson Farb and Howard Masur, Teichmüller geometry of moduli space. II. $\scr M(S)$ seen from far away, In the tradition of Ahlfors-Bers. V, Contemp. Math., vol. 510, Amer. Math. Soc., Providence, RI, 2010, pp. 71–79. MR 2581831, DOI 10.1090/conm/510/10019
Allen Hatcher, Algebraic topology, Cambridge University Press, Cambridge, 2002. MR 1867354
Steven P. Kerckhoff, The asymptotic geometry of Teichmüller space, Topology 19 (1980), no. 1, 23–41. MR 559474, DOI 10.1016/0040-9383(80)90029-4
Enrico Leuzinger, Reduction theory for mapping class groups and applications to moduli spaces, J. Reine Angew. Math. 649 (2010), 11–31. MR 2746464, DOI 10.1515/CRELLE.2010.086
Lixin Liu, Athanase Papadopoulos, Weixu Su, and Guillaume Théret, Length spectra and the Teichmüller metric for surfaces with boundary, Monatsh. Math. 161 (2010), no. 3, 295–311. MR 2726215, DOI 10.1007/s00605-009-0145-8
Lixin Liu, Athanase Papadopoulos, Weixu Su, and Guillaume Théret, On length spectrum metrics and weak metrics on Teichmüller spaces of surfaces with boundary, Ann. Acad. Sci. Fenn. Math. 35 (2010), no. 1, 255–274. MR 2643408, DOI 10.5186/aasfm.2010.3515
L. Liu and W. Su, Almost-isometry between Teichmüller metric and length-spectrum metric on moduli space, Bull. Lond. Math. Soc. 43 (2011), no. 6, 1181–1190. MR 2861539, DOI 10.1112/blms/bdr052
Bernard Maskit, Comparison of hyperbolic and extremal lengths, Ann. Acad. Sci. Fenn. Ser. A I Math. 10 (1985), 381–386. MR 802500, DOI 10.5186/aasfm.1985.1042
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
Maryam Mirzakhani, Simple geodesics and Weil-Petersson volumes of moduli spaces of bordered Riemann surfaces, Invent. Math. 167 (2007), no. 1, 179–222. MR 2264808, DOI 10.1007/s00222-006-0013-2
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, DOI 10.5186/aasfm.1975.0122
W. P. Thurston, A spine for Teichmüller space, preprint.