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Extremal length boundary of the Teichmüller space contains non-Busemann points


Author: Hideki Miyachi
Journal: Trans. Amer. Math. Soc. 366 (2014), 5409-5430
MSC (2010): Primary 30F60, 32G15, 30C75, 31B15; Secondary 30C62, 51F99
DOI: https://doi.org/10.1090/S0002-9947-2014-06145-X
Published electronically: May 21, 2014
MathSciNet review: 3240928
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Abstract: We present an overview of the extremal length embedding of a Teichmüller space and its extremal length compactification. For Teichmüller spaces of dimension at least two, we describe a large class of non-Busemann points on the metric boundary, that is, points that cannot be realized as limits of almost geodesic rays.


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  • [1] Lars V. Ahlfors, Lectures on quasiconformal mappings, Manuscript prepared with the assistance of Clifford J. Earle, Jr. Van Nostrand Mathematical Studies, No. 10, D. Van Nostrand Co., Inc., Toronto, Ont.-New York-London, 1966. MR 0200442 (34 #336)
  • [2] Lars V. Ahlfors, Conformal invariants: topics in geometric function theory, McGraw-Hill Book Co., New York, 1973. McGraw-Hill Series in Higher Mathematics. MR 0357743 (50 #10211)
  • [3] Marianne Akian, Stéphane Gaubert, and Cormac Walsh, The max-plus Martin boundary, Doc. Math. 14 (2009), 195-240. MR 2538616 (2011e:31015)
  • [4] Lipman Bers, Quasiconformal mappings and Teichmüller's theorem, Analytic functions, Princeton Univ. Press, Princeton, N.J., 1960, pp. 89-119. MR 0114898 (22 #5716)
  • [5] Martin R. Bridson and André Haefliger, Metric spaces of non-positive curvature, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 319, Springer-Verlag, Berlin, 1999. MR 1744486 (2000k:53038)
  • [6] Travaux de Thurston sur les surfaces, Société Mathématique de France, Paris, 1991 (French). Séminaire Orsay; Reprint of Travaux de Thurston sur les surfaces, Soc. Math.France, Paris, 1979 [MR0568308 (82m:57003)]; Astérisque No. 66-67 (1991). MR 1134426 (92g:57001)
  • [7] Clifford J. Earle, Reduced Teichmüller spaces, Trans. Amer. Math. Soc. 126 (1967), 54-63. MR 0204642 (34 #4481)
  • [8] Frederick P. Gardiner, Measured foliations and the minimal norm property for quadratic differentials, Acta Math. 152 (1984), no. 1-2, 57-76. MR 736212 (85i:30085), https://doi.org/10.1007/BF02392191
  • [9] Frederick P. Gardiner and Howard Masur, Extremal length geometry of Teichmüller space, Complex Variables Theory Appl. 16 (1991), no. 2-3, 209-237. MR 1099913 (92f:32034)
  • [10] M. Gromov, Hyperbolic manifolds, groups and actions, Brook Conference (State Univ. New York, Stony Brook, N.Y., 1978) Ann. of Math. Stud., vol. 97, Princeton Univ. Press, Princeton, N.J., 1981, pp. 183-213. MR 624814 (82m:53035)
  • [11] John Hubbard and Howard Masur, Quadratic differentials and foliations, Acta Math. 142 (1979), no. 3-4, 221-274. MR 523212 (80h:30047), https://doi.org/10.1007/BF02395062
  • [12] 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 (94b:32031)
  • [13] Vadim A. Kaimanovich and Howard Masur, The Poisson boundary of the mapping class group, Invent. Math. 125 (1996), no. 2, 221-264. MR 1395719 (97m:32033), https://doi.org/10.1007/s002220050074
  • [14] Vadim A. Kaimanovich and Howard Masur, The Poisson boundary of Teichmüller space, J. Funct. Anal. 156 (1998), no. 2, 301-332. MR 1636940 (99h:30044), https://doi.org/10.1006/jfan.1998.3252
  • [15] A. Karlsson, V. Metz, and G. A. Noskov, Horoballs in simplices and Minkowski spaces, Int. J. Math. Math. Sci. , posted on (2006), Art. ID 23656, 20. MR 2268510 (2007k:53047), https://doi.org/10.1155/IJMMS/2006/23656
  • [16] Steven P. Kerckhoff, The asymptotic geometry of Teichmüller space, Topology 19 (1980), no. 1, 23-41. MR 559474 (81f:32029), https://doi.org/10.1016/0040-9383(80)90029-4
  • [17] O. Lehto and K. I. Virtanen, Quasiconformal mappings in the plane, 2nd ed., Springer-Verlag, New York, 1973. Translated from the German by K. W. Lucas; Die Grundlehren der mathematischen Wissenschaften, Band 126. MR 0344463 (49 #9202)
  • [18] 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 (2011k:32017), https://doi.org/10.1007/s00605-009-0145-8
  • [19] L. Liu and W. Su, The horofunction compactification of Teichmüller metric, preprint, ArXiv.org : http://arxiv.org/abs/1012.0409.
  • [20] Albert Marden and Kurt Strebel, The heights theorem for quadratic differentials on Riemann surfaces, Acta Math. 153 (1984), no. 3-4, 153-211. MR 766263 (86a:30076), https://doi.org/10.1007/BF02392377
  • [21] Howard Masur, On a class of geodesics in Teichmüller space, Ann. of Math. (2) 102 (1975), no. 2, 205-221. MR 0385173 (52 #6038)
  • [22] Howard Masur, Random walks on Teichmuller space and the mapping class group, J. Anal. Math. 67 (1995), 117-164. MR 1383491 (97e:32022), https://doi.org/10.1007/BF02787787
  • [23] Howard A. Masur and Yair N. Minsky, Geometry of the complex of curves. I. Hyperbolicity, Invent. Math. 138 (1999), no. 1, 103-149. MR 1714338 (2000i:57027), https://doi.org/10.1007/s002220050343
  • [24] Albert Marden and Howard Masur, A foliation of Teichmüller space by twist invariant disks, Math. Scand. 36 (1975), no. 2, 211-228. MR 0393584 (52 #14393)
  • [25] Hideki Miyachi, On Gardiner-Masur boundary of Teichmüller space, Complex analysis and its applications, OCAMI Stud., vol. 2, Osaka Munic. Univ. Press, Osaka, 2007, pp. 295-300. MR 2404935 (2009e:32013)
  • [26] Hideki Miyachi, Teichmüller rays and the Gardiner-Masur boundary of Teichmüller space, Geom. Dedicata 137 (2008), 113-141. MR 2449148 (2009m:30093), https://doi.org/10.1007/s10711-008-9289-2
  • [27] Hideki Miyachi, Teichmüller rays and the Gardiner-Masur boundary of Teichmüller space II, Geom. Dedicata 162 (2013), 283-304. MR 3009545, https://doi.org/10.1007/s10711-012-9727-z
  • [28] H. Miyachi, Unification of extremal length geometry on Teichmüller space via intersection number, submitted.
  • [29] Yair N. Minsky, Teichmüller geodesics and ends of hyperbolic $ 3$-manifolds, Topology 32 (1993), no. 3, 625-647. MR 1231968 (95g:57031), https://doi.org/10.1016/0040-9383(93)90013-L
  • [30] Yair N. Minsky, Extremal length estimates and product regions in Teichmüller space, Duke Math. J. 83 (1996), no. 2, 249-286. MR 1390649 (97b:32019), https://doi.org/10.1215/S0012-7094-96-08310-6
  • [31] A. Papadopoulos, Problem 13 in Problem Session Teichmüller Theory, Oberwolfach Reports Vol. 7, (2010) Issue 4, 3085-3157.
  • [32] Marc A. Rieffel, Group $ C^*$-algebras as compact quantum metric spaces, Doc. Math. 7 (2002), 605-651 (electronic). MR 2015055 (2004k:22009)
  • [33] Kurt Strebel, Quadratic differentials, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], vol. 5, Springer-Verlag, Berlin, 1984. MR 743423 (86a:30072)
  • [34] Cormac Walsh, The horofunction boundary of the Hilbert geometry, Adv. Geom. 8 (2008), no. 4, 503-529. MR 2456635 (2009m:53108), https://doi.org/10.1515/ADVGEOM.2008.032
  • [35] C. Walsh, The horoboundary and isometry group of Thurson's Lipschitz metric, preprint, ArXiv.org : http://arxiv.org/abs/1006.2158
  • [36] C. Walsh, The asymptotic geometry of the Teichmüller metric, preprint. (2012).
  • [37] Corran Webster and Adam Winchester, Busemann points of infinite graphs, Trans. Amer. Math. Soc. 358 (2006), no. 9, 4209-4224 (electronic). MR 2219016 (2007d:46059), https://doi.org/10.1090/S0002-9947-06-03877-3

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Additional Information

Hideki Miyachi
Affiliation: Department of Mathematics, Graduate School of Science, Osaka University, Machikaneyama 1-1, Toyonaka, Osaka 560-0043, Japan

DOI: https://doi.org/10.1090/S0002-9947-2014-06145-X
Keywords: Teichm\"uller space, Teichm\"uller distance, extremal length, metric boundary, horofunction boundary, Busemann point
Received by editor(s): March 31, 2012
Received by editor(s) in revised form: August 23, 2012, and December 8, 2012
Published electronically: May 21, 2014
Additional Notes: The author was partially supported by the Ministry of Education, Science, Sports and Culture, Grant-in-Aid for Scientific Research (C), 21540177
Article copyright: © Copyright 2014 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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