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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Metric properties of the group of area preserving diffeomorphisms


Authors: Michel Benaim and Jean-Marc Gambaudo
Journal: Trans. Amer. Math. Soc. 353 (2001), 4661-4672
MSC (1991): Primary 20F36, 58B05, 58B25, 76A02
Published electronically: June 14, 2001
MathSciNet review: 1851187
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Abstract:

Area preserving diffeomorphisms of the 2-disk which are identity near the boundary form a group ${\mathcal D}_2$ which can be equipped, using the $L^2$-norm on its Lie algebra, with a right invariant metric. With this metric the diameter of ${\mathcal D}_2$ is infinite. In this paper we show that ${\mathcal D}_2$ contains quasi-isometric embeddings of any finitely generated free group and any finitely generated abelian free group.


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Additional Information

Michel Benaim
Affiliation: Université de Cergy Pontoise, Laboratoire de Mathématiques, 2, avenue Adolphe Chauvin, 95302 Cergy-Pontoise Cedex, France
Email: benaim@math.u-cergy.fr

Jean-Marc Gambaudo
Affiliation: Université de Bourgogne, Laboratoire de Topologie, UMR CNRS 5584, B.P. 47870-21078-Dijon Cedex, France

DOI: http://dx.doi.org/10.1090/S0002-9947-01-02808-2
PII: S 0002-9947(01)02808-2
Keywords: Area preserving diffeomorphisms, braids, free groups, quasi-isometry
Received by editor(s): April 11, 2000
Received by editor(s) in revised form: October 30, 2000
Published electronically: June 14, 2001
Article copyright: © Copyright 2001 American Mathematical Society