Commutators and diffeomorphisms of surfaces. (English) Zbl 1088.37018

Gambaudo, Jean-Marc; Ghys, Étienne

Ergodic Theory Dyn. Syst. 24, No. 5, 1591-1617 (2004).

Summary: For any compact oriented surface \(\Sigma\), we consider the group of diffeomorphisms of \(\Sigma\) which preserve a given area form. 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.

MSC:

37E30	Dynamical systems involving homeomorphisms and diffeomorphisms of planes and surfaces
37C05	Dynamical systems involving smooth mappings and diffeomorphisms
57M60	Group actions on manifolds and cell complexes in low dimensions
20F28	Automorphism groups of groups

Keywords:

area-preserving diffeomorphisms; quasi-morphisms