Extremal length boundary of the Teichmüller space contains non-Busemann points
HTML articles powered by AMS MathViewer
- by Hideki Miyachi PDF
- Trans. Amer. Math. Soc. 366 (2014), 5409-5430 Request permission
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.References
- Lars V. Ahlfors, Lectures on quasiconformal mappings, Van Nostrand Mathematical Studies, No. 10, D. Van Nostrand Co., Inc., Toronto, Ont.-New York-London, 1966. Manuscript prepared with the assistance of Clifford J. Earle, Jr. MR 0200442
- Lars V. Ahlfors, Conformal invariants: topics in geometric function theory, McGraw-Hill Series in Higher Mathematics, McGraw-Hill Book Co., New York-Düsseldorf-Johannesburg, 1973. MR 0357743
- Marianne Akian, Stéphane Gaubert, and Cormac Walsh, The max-plus Martin boundary, Doc. Math. 14 (2009), 195–240. MR 2538616
- Lipman Bers, Quasiconformal mappings and Teichmüller’s theorem, Analytic functions, Princeton Univ. Press, Princeton, N.J., 1960, pp. 89–119. MR 0114898
- 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, DOI 10.1007/978-3-662-12494-9
- 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) (1991). MR 1134426
- Clifford J. Earle, Reduced Teichmüller spaces, Trans. Amer. Math. Soc. 126 (1967), 54–63. MR 204642, DOI 10.1090/S0002-9947-1967-0204642-3
- Frederick P. Gardiner, Measured foliations and the minimal norm property for quadratic differentials, Acta Math. 152 (1984), no. 1-2, 57–76. MR 736212, DOI 10.1007/BF02392191
- 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, DOI 10.1080/17476939108814480
- M. Gromov, Hyperbolic manifolds, groups and actions, Riemann surfaces and related topics: Proceedings of the 1978 Stony 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
- John Hubbard and Howard Masur, Quadratic differentials and foliations, Acta Math. 142 (1979), no. 3-4, 221–274. MR 523212, DOI 10.1007/BF02395062
- 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, DOI 10.1007/978-4-431-68174-8
- Vadim A. Kaimanovich and Howard Masur, The Poisson boundary of the mapping class group, Invent. Math. 125 (1996), no. 2, 221–264. MR 1395719, DOI 10.1007/s002220050074
- Vadim A. Kaimanovich and Howard Masur, The Poisson boundary of Teichmüller space, J. Funct. Anal. 156 (1998), no. 2, 301–332. MR 1636940, DOI 10.1006/jfan.1998.3252
- 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, DOI 10.1155/IJMMS/2006/23656
- 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
- O. Lehto and K. I. Virtanen, Quasiconformal mappings in the plane, 2nd ed., Die Grundlehren der mathematischen Wissenschaften, Band 126, Springer-Verlag, New York-Heidelberg, 1973. Translated from the German by K. W. Lucas. MR 0344463, DOI 10.1007/978-3-642-65513-5
- 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
- L. Liu and W. Su, The horofunction compactification of Teichmüller metric, preprint, ArXiv.org : http://arxiv.org/abs/1012.0409.
- 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, DOI 10.1007/BF02392377
- Howard Masur, On a class of geodesics in Teichmüller space, Ann. of Math. (2) 102 (1975), no. 2, 205–221. MR 385173, DOI 10.2307/1971031
- Howard Masur, Random walks on Teichmuller space and the mapping class group, J. Anal. Math. 67 (1995), 117–164. MR 1383491, DOI 10.1007/BF02787787
- 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, DOI 10.1007/s002220050343
- Albert Marden and Howard Masur, A foliation of Teichmüller space by twist invariant disks, Math. Scand. 36 (1975), no. 2, 211–228. MR 393584, DOI 10.7146/math.scand.a-11572
- 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
- Hideki Miyachi, Teichmüller rays and the Gardiner-Masur boundary of Teichmüller space, Geom. Dedicata 137 (2008), 113–141. MR 2449148, DOI 10.1007/s10711-008-9289-2
- Hideki Miyachi, Teichmüller rays and the Gardiner-Masur boundary of Teichmüller space II, Geom. Dedicata 162 (2013), 283–304. MR 3009545, DOI 10.1007/s10711-012-9727-z
- H. Miyachi, Unification of extremal length geometry on Teichmüller space via intersection number, submitted.
- Yair N. Minsky, Teichmüller geodesics and ends of hyperbolic $3$-manifolds, Topology 32 (1993), no. 3, 625–647. MR 1231968, DOI 10.1016/0040-9383(93)90013-L
- 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
- A. Papadopoulos, Problem 13 in Problem Session Teichmüller Theory, Oberwolfach Reports Vol. 7, (2010) Issue 4, 3085–3157.
- Marc A. Rieffel, Group $C^*$-algebras as compact quantum metric spaces, Doc. Math. 7 (2002), 605–651. MR 2015055
- 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, DOI 10.1007/978-3-662-02414-0
- Cormac Walsh, The horofunction boundary of the Hilbert geometry, Adv. Geom. 8 (2008), no. 4, 503–529. MR 2456635, DOI 10.1515/ADVGEOM.2008.032
- C. Walsh, The horoboundary and isometry group of Thurson’s Lipschitz metric, preprint, ArXiv.org : http://arxiv.org/abs/1006.2158
- C. Walsh, The asymptotic geometry of the Teichmüller metric, preprint. (2012).
- Corran Webster and Adam Winchester, Busemann points of infinite graphs, Trans. Amer. Math. Soc. 358 (2006), no. 9, 4209–4224. MR 2219016, DOI 10.1090/S0002-9947-06-03877-3
Additional Information
- Hideki Miyachi
- Affiliation: Department of Mathematics, Graduate School of Science, Osaka University, Machikaneyama 1-1, Toyonaka, Osaka 560-0043, Japan
- MR Author ID: 650573
- 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
- © Copyright 2014
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - 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
- MathSciNet review: 3240928