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On the topology defined by Thurston's asymmetric metric

Published online by Cambridge University Press:  01 May 2007

ATHANASE PAPADOPOULOS  and
GUILLAUME THÉRET
ATHANASE PAPADOPOULOS
Affiliation:
Institut de Recherche Mathématique Avancée, Université Louis Pasteur and CNRS, 7 rue René Descartes, 67084 Strasbourg Cedex - France. e-mail: papadopoulos@math.u-strasbg.fr, theret@math.u-strasbg.fr
GUILLAUME THÉRET
Affiliation:
Institut de Recherche Mathématique Avancée, Université Louis Pasteur and CNRS, 7 rue René Descartes, 67084 Strasbourg Cedex - France. e-mail: papadopoulos@math.u-strasbg.fr, theret@math.u-strasbg.fr
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Abstract

We establish some properties of Thurston's asymmetric metric L on the Teichmüller space of a surface of genus with punctures and with negative Euler characteristic. We study convergence of sequences of elements in in the sense of L, as well as sequences that tend to infinity in . We show that the topology that the asymmetric metric L induces on Teichmüller space is the same as the usual topology. Furthermore, we show that L satisfies the axioms of a (not necessarily symmetric) metric in the sense of Busemann and conclude that L is complete in the sense of Busemann.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 142 , Issue 3 , May 2007 , pp. 487 - 496
Copyright
Copyright © Cambridge Philosophical Society 2007

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References

REFERENCES

[1]Abikoff, W.. The Real Analytic Theory of Teichmüller Space. Lecture Notes in Math. 820 (Springer-Verlag, 1980).Google Scholar
[2]Busemann, H.. Recent synthetic differential geometry. Ergeb. Math. Grenzgeb. 54 (1970).Google Scholar
[3]Fathi, A., Laudenbach, F. and Poénaru, V.. Travaux de Thurston sur les surfaces. Astérisque 66–67 (1979).Google Scholar
[4]Imayoshi, Y. and Taniguchi, M.. An Introduction to Teichmüller Spaces (Springer-Verlag, 1992).CrossRefGoogle Scholar
[5]Papadopoulos, A.. On Thurston's boundary of Teichmüller space and the extension of earthquakes. Topology Appl. 41 (1991), no. 3, 147–177.Google Scholar
[6]Penner, R. C. and Harer, J.. Combinatorics of train tracks. Annals of Math. Studies 125 (1992).Google Scholar
[7]Thurston, W. P.. The Geometry and Topology of Three-Manifolds. Mimeographed notes (Princeton University, 1976).Google Scholar
[8]Thurston, W. P.. Minimal stretch maps between hyperbolic surfaces, preprint (1986).Google Scholar
[9]Thurston, W. P.. Three-Dimensional Geometry and Topology. Vol. 1, edited by Levy, Silvio. Princeton Mathematical Series 35 (Princeton University Press, 1997).CrossRefGoogle Scholar