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Commutator length of annulus diffeomorphisms

Published online by Cambridge University Press:  11 December 2012

E. MILITON
E. MILITON*
Affiliation:
Laboratoire de Mathématiques d’Orsay, UMR 8628, Bât. 425, Faculté des Sciences d’Orsay, Université Paris-Sud XI, F-91405 Orsay cedex, France (email: [email protected])
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Abstract

We study the group Diffr0(𝔸) of Cr-diffeomorphisms of the closed annulus that are isotopic to the identity. We show that, for r≠2,3, the linear space of homogeneous quasi-morphisms on the group Diffr0(𝔸) is one-dimensional. Therefore, the commutator length on this group is (stably) unbounded. In particular, this provides an example of a manifold whose diffeomorphism group is unbounded in the sense of Burago, Ivanov and Polterovich.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 34 , Issue 3 , June 2014 , pp. 919 - 937
Copyright
©2012 Cambridge University Press 

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References

[1]Abe, K. and Fukui, K.. Commutators of C1-diffeomorphisms preserving a submanifold. J. Math. Soc. Japan 61 (2009), 427–436.Google Scholar
[2]Anderson, R. D.. The algebraic simplicity of certain groups of homeomorphisms. Amer. J. Math. 80(4) (Oct. 1958), 955–963.CrossRefGoogle Scholar
[3]Bavard, C.. Longueur stable des commutateurs. Enseign. Math (2) 37(1–2) (1991), 109–150.Google Scholar
[4]Bounemoura, A.. Simplicité des groupes de transformations de surface (Ensaios Matematicos, 14). Sociedade Brasileira de Matematica, Rio de Janeiro, 2008, pp. 1–143.Google Scholar
[5]Burago, D., Ivanov, S. and Polterovich, L.. Conjugation-invariant norms on groups of geometric origin. Groups of diffeomorphisms: In Honor of Shigeyuki Morita on the Occasion of His 60th Birthday (Advanced Studies in Pure Mathematics, 52). Mathematical Society of Japan, Tokyo, 2008.Google Scholar
[6]Calegari, D.. scl (Mathematical Society of Japan Memoirs, 20). Mathematical Society of Japan, Tokyo, 2009.CrossRefGoogle Scholar
[7]Ghys, E.. Groups acting on the circle. Enseign. Math. (2) 47 (2001), 329–407.Google Scholar
[8]Eisenbud, D., Hirsch, U. and Neumann, W.. Transverse foliations of Seifert bundles and self-homeomorphisms of the circle. Comment. Math. Helv. 56 (1981), 638–660.CrossRefGoogle Scholar
[9]Entov, M., Polterovich, L. and Py, P.. On continuity of quasi-morphisms for symplectic maps. Perspectives in Analysis, Geometry and Topology (Progress in Mathematics, 296). Birkhäuser/Springer, New York, 2012, pp. 169–197.Google Scholar
[10]Mather, J. N.. Commutators of diffeomorphisms I. Comment. Math. Helv. 49 (1974), 512–528.Google Scholar
[11]Mather, J. N.. Commutators of diffeomorphisms II. Comment. Math. Helv. 50 (1975), 33–40.Google Scholar
[12]Tsuboi, T.. On the uniform perfectness of diffeomorphism groups. Groups of Diffeomorphisms: In Honor of Shigeyuki Morita on the Occasion of His 60th Birthday (Advanced Studies in Pure Mathematics, 52). Mathematical Society of Japan, Tokyo, 2008.Google Scholar