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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
DOI: https://doi.org/10.1090/S0002-9947-01-02808-2
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. ARNOLD, V. AND KHESIN, B. Topological methods in hydrodynamics. Applied Mathematical Sciences, 125, (1998). MR 99b:58002
  • 2. BANYAGA, A.: On the group of diffeomorphisms preserving an exact symplectic form, in Differential topology, Varenna (1976), 5-9. MR 83g:58006
  • 3. BIRMAN, J. Braids, Links and Mapping Class Groups. Annals of Math. Studies 82, Princeton University Press, (1974) Erratum, 1975. MR 51:11477; MR 54:13894
  • 4. CALABI, E.: On the group of automorphisms of a symplectic manifold, in Problems in analysis, Symposium in honor of S. Bochner, R. C. Gunning, Ed., Princeton Univ. Press, Princeton (1970), 1-26. MR 50:3268
  • 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. ELIASHBERG, Y. AND RATIU, T.: The diameter of the symplectomorphism group is infinite, Invent. Math. 103, (1991), 327-340. MR 92a:58018
  • 7. FATHI, A.: Transformations et homéomorphismes préservant la mesure. Systèmes dynamiques minimaux., Thèse Orsay (1980).
  • 8. GAMBAUDO, J.-M. AND GHYS, ´E.: Enlacements asymptotiques, Topology 36, (1997), 1355-1379. MR 98f:57050
  • 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. GHYS, ´E. AND DE LA HARPE, P. Sur les groupes hyperboliques d'après Mikhael Gromov, Ghys, É and de la Harpe, eds., Progress in Mathematics 83, Birkhauser, (1990). MR 92f:53050
  • 11. KINGMAN, J. F. C.: The ergodic theory of subadditive stochastic processes, J. Royal Stat. Soc., 30, (1968), 499-510. MR 40:8114
  • 12. MURASUGI, K. Knot theory and its applications. Translated from the 1993 Japanese original by Bohdan Kurpita. Birkhäuser Boston, Inc., Boston, (1996). MR 97g:57011
  • 13. SHNIRELMAN, A.: The geometry of the group of diffeomorphisms and the dynamics of an ideal incompressible fluid, Matem. Sbornik 128, (1985), 82-109; English transl: Math. USSR, Sbornik 56 (1987), 79-105. MR 87d:58034

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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: https://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

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