Hostname: page-component-586b7cd67f-l7hp2 Total loading time: 0 Render date: 2024-11-30T19:38:41.468Z Has data issue: false hasContentIssue false

A characterization of biholomorphic automorphisms of Teichmüller space

Published online by Cambridge University Press:  30 August 2012

RYOSUKE MINEYAMA
Affiliation:
Department of Mathematics, Graduate School of Science, Osaka University, Machikaneyama 1-1, Toyonaka, Osaka 560-0043, Japan. e-mail: [email protected], [email protected]
HIDEKI MIYACHI
Affiliation:
Department of Mathematics, Graduate School of Science, Osaka University, Machikaneyama 1-1, Toyonaka, Osaka 560-0043, Japan. e-mail: [email protected], [email protected]

Abstract

In this paper, we give an alternative approach to Royden–Earle–Kra–Markovic's characterization of biholomorphic automorphisms of Teichmüller space of Riemann surface of analytically finite type.

Type
Research Article
Copyright
Copyright © Cambridge Philosophical Society 2012

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

[1]Abikoff, W.The real analytic theory of Teichmüller space. Lecture notes in Math. Series 820 (Springer-Verlag, NewYork, 1989).Google Scholar
[2]Dehn, M.. Stillwell, J. (Ed), Papers on Group Theory and Topology (Springer, Berlin, 1987).CrossRefGoogle Scholar
[3]Earle, C. and Gardiner, F.Geometric isomorphisms between infinite dimensional Teichmüller spaces. Trans. Amer. Math. Soc. 348 (1996), 11631190.CrossRefGoogle Scholar
[4]Earle, C. and Kra, I.On isometries between Teichmüller spaces. Duke Math. J. 41 (1974), 583591.CrossRefGoogle Scholar
[5]Earle, C. and Markovic, V.Isometries between the spaces of L 1 holomorphic quadratic differentials on Riemann surfaces of finite type. Duke Math. J. 120 (2003), no. 2, 433440.CrossRefGoogle Scholar
[6]Douady, A., Fathi, A., Fried, D., Laudenbach, F., Poénaru, V. and Shub, M.Travaux de Thurston sur les surfaces (Séminaire Orsay (seconde édition) Astérisque No. 66-67 (Société Mathématique de France, Paris, 1991).Google Scholar
[7]Fletcher, A. and Markovic, V.Quasiconformal Maps and Teichmüller Theory. Oxford Graduate Texts in Mathematics (2006).CrossRefGoogle Scholar
[8]Gardiner, F.Teichmüller Theory and Quadratic Differentials (Wiley-Interscience, 1987).Google Scholar
[9]Gardiner, F. and Masur, H.Extremal length geometry of Teichmüller space. Complex Variables Theory Appl. 16 (1991), no. 2-3, 209237.Google Scholar
[10]Hubbard, J. and Masur, H.Quadratic differentials and foliations. Acta Math. 142 (1979), no. 3-4, 221274.CrossRefGoogle Scholar
[11]Imayoshi, Y. and Taniguchi, M.Introduction to Teichmüller Spaces (Springer-Verlag, NewYork, 1992).CrossRefGoogle Scholar
[12]Ivanov, N.Isometries of Teichmüller spaces from the point of view of Mostow rigidity, Topology, Ergodic Theory, Real Algebraic Geometry (eds. Turaev, V., Vershik, A.). Amer. Math. Soc. Transl. Ser. 2, vol 202 (American Mathematical Society, 2001), 131149.Google Scholar
[13]Kerckhoff, S.The asymptotic geometry of Teichmüller space. Topology 19 (1980), 2341.CrossRefGoogle Scholar
[14]Korkmaz, M.Automorphisms of complexes of curves on punctured spheres and punctured tori. Topology and its Applications 95 (1999), 85111.CrossRefGoogle Scholar
[15]Kra, I.Horocyclic coordinates for Riemann surfaces and moduli spaces. I, Teichmüller and Riemann spaces of Kleinian groups. J. Amer. Math. Soc. 3 (1990), no. 3, 499578.Google Scholar
[16]Liu, L. and Su, W.The horofunction compactification of Teichmüller metric. To appear in Handbook of Teichmüller Theory (Papadopoulos, A., ed.), Volume IV, (EMS Publishing House, Zürich, 2012).Google Scholar
[17]Luo, F.Automorphisms of the complexes and curves. Topology 39 (2000), 283298.CrossRefGoogle Scholar
[18]Marden, A.The geometry of finitely generated Kleinian groups. Ann. of Math. 99 (1974), 383462.CrossRefGoogle Scholar
[19]Maskit, B.Kleinian groups. Grundlehren der Mathematischen Wissenschaften, 287 (Springer-Verlag, Berlin, 1988).Google Scholar
[20]Masur, H.On a class of geodesics in Teichmüller space. Ann. of Math. 102 (1975), 205221.CrossRefGoogle Scholar
[21]Masur, H.Interval exchange transformations and measured foliations. Ann. of Math. 115 (1982), 169200.CrossRefGoogle Scholar
[22]Masur, H. and Wolf, M.The Weil–Peterson isometry group. Geom. Dedicata 93 (2002), 177190.CrossRefGoogle Scholar
[23]Markovic, V.Biholomorphic maps between Teichmu Íller spaces. Duke Math. J. 120 (2003), 405431.CrossRefGoogle Scholar
[24]Minsky, Y.The classification of punctured-torus groups. Ann. of Math.J. 149 (1999), 559626.CrossRefGoogle Scholar
[25]Miyachi, H. On the Gardiner–Masur boundary of Teichmüller space, Proceedings of the 15th ICFIDCAA (Osaka, 2007), OCAMI Studies 2 (2008), 295–300.Google Scholar
[26]Miyachi, H.Teichmüller rays and the Gardiner–Masur boundary of Teichmüller space, Geom. Dedicata 137 (2008), 113141.CrossRefGoogle Scholar
[27]Miyachi, H. Teichmüller rays and the Gardiner-Masur boundary of Teichmüller space II. To appear in Geom. Dedicata.Google Scholar
[28]Miyachi, H. Unification of the extremal length geometry on Teichmüller space via intersection number, submitted.Google Scholar
[29]Royden, H.Automorphisms and isometries of Teichmüller space, Advances in the Theory of Riemann Surfaces. Ann. of Math. Studies 66 (Princeton University Press, Princeton, N.J., 1971), pp. 369383.Google Scholar
[30]Shiga, H. and Tanigawa, H.On the Maskit coordinates of Teichmüller spaces and modular transformations. Kodai Math. J. 12 (1989), 437443.CrossRefGoogle Scholar
[31]Matsuzaki, K. and Taniguchi, M.Hyperbolic manifolds and Kleinian groups. Oxford Mathematical Monographs. Oxford Science Publications (The Clarendon Press, Oxford University Press, New York, 1998).CrossRefGoogle Scholar