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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

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