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On the classification of mapping class actions on Thurston's asymmetric metric

Published online by Cambridge University Press:  28 June 2013

L. LIU ,
A. PAPADOPOULOS ,
W. SU  and
G. THÉRET
L. LIU
Affiliation:
Department of Mathematics, Sun Yat–Sen University, 510275, Guangzhou, P. R. China. e-mail: mcsllx@mail.sysu.edu.cn
A. PAPADOPOULOS
Affiliation:
Université de Strasbourg and CNRS, 7 Rue René Descartes, 67084 Strasbourg Cedex, France. e-mail: athanase.papadopoulos@math.unistra.fr
W. SU
Affiliation:
Department of Mathematics, Fudan University, 200433, Shanghai, P. R. China, and Université de Strasbourg and CNRS, 7 Rue René Descartes, 67084 Strasbourg Cedex, France. e-mail: suweixu@gmail.com
G. THÉRET
Affiliation:
Institut de Mathématiques de Bourgogne, Université de Bourgogne UMR 5584 du CNRSBP 47870 21078 Dijon Cedex, France. e-mail: guillaume.theret71@orange.fr
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Abstract

We study the action of the elements of the mapping class group of a surface of finite type on the Teichmüller space of that surface equipped with Thurston's asymmetric metric. We classify such actions as elliptic, parabolic, hyperbolic and pseudo-hyperbolic, depending on whether the translation distance of such an element is zero or positive and whether the value of this translation distance is attained or not, and we relate these four types to Thurston's classification of mapping class elements. The study is parallel to the one made by Bers in the setting of Teichmüller space equipped with Teichmüller's metric, and to the one made by Daskalopoulos and Wentworth in the setting of Teichmüller space equipped with the Weil–Petersson metric.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 155 , Issue 3 , November 2013 , pp. 499 - 515
Copyright
Copyright © Cambridge Philosophical Society 2013 

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References

REFERENCES

[1]Bers, L.An extremal problem for quasiconformal mappings and a theorem by Thurston. Acta Math. 141, No. 1 (1978), 73–98.CrossRefGoogle Scholar
[2]Bers, L.A remark on Mumford's compactness theorem. Israel J. Math. 12, No. 4 (1972), 400–407.CrossRefGoogle Scholar
[3]Casson, A. J. and Bleiler, S. A.Automorphisms of surfaces after Nielsen and Thurston. London Math. Soc. Student Texts, 9 (Cambridge University Press, Cambridge, 1988).CrossRefGoogle Scholar
[4]Buser, P.Geometry and spectra of compact Riemann surfaces (Birkhäuser, 1992).Google Scholar
[5]Daskalopoulos, G. and Wentworth, R.Classification of Weil–Petersson isometries. Amer. J. Math. 125, No. 4 (2003), 941–975.CrossRefGoogle Scholar
[6]Fathi, A., Laudenbach, F. and Poénaru, V. Travaux de Thurston sur les surfaces (second edition). Astérisque 66–67.Google Scholar
[7]Handel, M. and Thurston, W. P.New proofs of some results of Nielsen. Adv. Math. 56 (1985), 173–191.CrossRefGoogle Scholar
[8]Li, Z.Teichmüller metric and the length spectrums of Riemann surfaces. Scientia Sinica Ser. A 24 (1986), 265–274.Google Scholar
[9]Liu, L.On the metrics of length spectrum in Teichmüller space. Chin. J. Contemp. Math. 22, No. 1 (2001), 23–34.Google Scholar
[10]Masur, H.Two boundaries of Teichmüller space. Duke Math. 49, No. 1 (1982), 183–190.CrossRefGoogle Scholar
[11]Nielsen, J.Abbildungklassen Endlicher Ordung. Acta Math. 75 (1943), 23–115.CrossRefGoogle Scholar
[12]Papadopoulos, A.On Thurston's boundary of Teichmüller space and the extension of earthquakes. Topology Appl. 41 (1991), 147–177.CrossRefGoogle Scholar
[13]Papadopoulos, A. and Théret, G.On the topology defined by Thurston's asymmetric metric. Math. Proc. Camb. Phil. Soc. 142 (2007), 487–496.CrossRefGoogle Scholar
[14]Papadopoulos, A. and Théret, G.On Teichmüller's metric and Thurston's asymmetric metric on Teichmüller space. In Papadopoulos, A. (ed.), Handbook of Teichmüller Theory. Volume I. Zürich, European Mathematical Society, IRMA Lectures in Mathematics and Theoretical Physics 11 (2007), 111–204.CrossRefGoogle Scholar
[15]Théret, G.On the negative convergence of Thurston's stretch lines towards the boundary of Teichmüller space. Ann. Acad. Sci. Fenn. Math. 32 (2007), 381–408.Google Scholar
[16]Théret, G. À propos de la convergence des lignes d'étirement vers le bord de Thurston de l'espace de Teichmüller. PhD thesis (University of Strasbourg, 2005).Google Scholar
[17]Thurston, W. P.On the geometry and dynamics of diffeomorphisms of surfaces. Bull. Amer. Math. Soc. (N.S.) 19 (2) (1988), 417–431.CrossRefGoogle Scholar
[18]Thurston, W. P. Minimal stretch maps between hyperbolic surfaces. arxiv:math.GT/9801039.Google Scholar
[19]Wolpert, S.The length spectra as moduli for compact Riemann surfaces. Ann. of Math. 109 (1979), 323–351.CrossRefGoogle Scholar