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Commutators and diffeomorphisms of surfaces

Published online by Cambridge University Press:  18 October 2004

JEAN-MARC GAMBAUDO  and
ÉTIENNE GHYS
JEAN-MARC GAMBAUDO
Affiliation:
Université de Bourgogne, Institut de Mathématiques de Bourgogne, U.M.R. 5584 du CNRS, 9, Avenue Alain Savary, 21078 Dijon Cedex, France (e-mail: [email protected])
ÉTIENNE GHYS
Affiliation:
Unité de Mathématiques Pures et Appliquées de l'École Normale Supérieure de Lyon, U.M.R. 5669 du CNRS, 46, Allée d'Italie, 69364 Lyon Cedex 07, France (e-mail: [email protected])
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Abstract

For any compact oriented surface $\Sigma$ we consider the group of diffeomorphisms of $\Sigma$ which preserve a given area form. In this paper we show that the vector space of homogeneous quasi-morphisms on this group has infinite dimension. This result is proved by constructing explicitly and for each surface an infinite family of independent homogeneous quasi-morphisms. These constructions use simple arguments related to linking properties of the orbits of the diffeomorphisms.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 24 , Issue 5 , October 2004 , pp. 1591 - 1617
Copyright
© 2004 Cambridge University Press

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