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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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  • 1. Vladimir I. Arnold and Boris A. Khesin, Topological methods in hydrodynamics, Applied Mathematical Sciences, vol. 125, Springer-Verlag, New York, 1998. MR 1612569
  • 2. A. Banyaga, On the group of diffeomorphisms preserving an exact symplectic form, Differential topology (Varenna, 1976) Liguori, Naples, 1979, pp. 5–9. MR 660656
  • 3. Joan S. Birman, Braids, links, and mapping class groups, Princeton University Press, Princeton, N.J.; University of Tokyo Press, Tokyo, 1974. Annals of Mathematics Studies, No. 82. MR 0375281
    Joan S. Birman, Erratum: “Braids, links, and mapping class groups” (Ann. of Math. Studies, No. 82, Princeton Univ. Press, Princeton, N. J., 1974), Princeton University Press, Princeton, N. J.; University of Tokyo Press, Toyko, 1975. Based on lecture notes by James Cannon. MR 0425944
  • 4. Eugenio Calabi, On the group of automorphisms of a symplectic manifold, Problems in analysis (Lectures at the Sympos. in honor of Salomon Bochner, Princeton Univ., Princeton, N.J., 1969) Princeton Univ. Press, Princeton, N.J., 1970, pp. 1–26. MR 0350776
  • 5. EBIN, D. G. AND MARSDEN, J.: Groups of diffeomorphisms and the notion of an incompressible fluid, Ann. of Math. 92, (1970), 102-163.
  • 6. Yakov Eliashberg and Tudor Ratiu, The diameter of the symplectomorphism group is infinite, Invent. Math. 103 (1991), no. 2, 327–340. MR 1085110, 10.1007/BF01239516
  • 7. FATHI, A.: Transformations et homéomorphismes préservant la mesure. Systèmes dynamiques minimaux., Thèse Orsay (1980).
  • 8. Jean-Marc Gambaudo and Étienne Ghys, Enlacements asymptotiques, Topology 36 (1997), no. 6, 1355–1379 (French). MR 1452855, 10.1016/S0040-9383(97)00001-3
  • 9. GAMBAUDO, J.-M. AND LAGRANGE, M.: Topological lower bounds on the distance between area preserving diffeomorphisms, Bol. Soc. Brasil. Mat. 31, (2000), 1-19. CMP 2000:12
  • 10. É. Ghys and P. de la Harpe (eds.), Sur les groupes hyperboliques d’après Mikhael Gromov, Progress in Mathematics, vol. 83, Birkhäuser Boston, Inc., Boston, MA, 1990 (French). Papers from the Swiss Seminar on Hyperbolic Groups held in Bern, 1988. MR 1086648
  • 11. J. F. C. Kingman, The ergodic theory of subadditive stochastic processes, J. Roy. Statist. Soc. Ser. B 30 (1968), 499–510. MR 0254907
  • 12. Kunio Murasugi, Knot theory and its applications, Birkhäuser Boston, Inc., Boston, MA, 1996. Translated from the 1993 Japanese original by Bohdan Kurpita. MR 1391727
  • 13. A. I. Shnirel′man, The geometry of the group of diffeomorphisms and the dynamics of an ideal incompressible fluid, Mat. Sb. (N.S.) 128(170) (1985), no. 1, 82–109, 144 (Russian). MR 805697

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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
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