Journal ArticleDOI
Commutators and diffeomorphisms of surfaces
Jean-Marc Gambaudo,Étienne Ghys +1 more
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In this paper, it was shown that the vector space of homogeneous quasi-morphisms on a group of diffeomorphisms preserves a given area form, and that for any compact oriented surfacewe, an infinite family of independent homogeneous quasimorphisms can be constructed explicitly for each surface.Abstract:
For any compact oriented surfacewe consider the group of diffeomorphisms ofwhich 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.read more
Citations
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Journal ArticleDOI
Right-veering diffeomorphisms of compact surfaces with boundary I
TL;DR: In this article, a detailed study of the monoid of right-veering diffeomorphisms on a compact oriented surface with nonempty boundary was conducted, in particular for the case when the surface is a punctured torus.
BookDOI
Knots and dynamics
TL;DR: In this article, the authors survey some advances in qualitative and quantitative description of this kind of phenomenon, including vorticity, helicity, and asymmetric cycles for flows and various notions of rotation and spin for surface diffeomorphisms.
BookDOI
An invitation to bounded cohomology
TL;DR: A selection of aspects of the theory of bounded cohomology is presented in this paper, with an emphasis on questions motivating the use of that theory as well as on some concrete issues suggested by its study.
Journal ArticleDOI
Maximal Representations of Surface Groups: Symplectic Anosov Structures
TL;DR: In this paper, a survey of maximal representations of homomorphic Lie groups is presented, which is the subset of Hom(Γg,G ) characterized by the maximality of the Toledo invariant.
Journal ArticleDOI
Right-veering diffeomorphisms of compact surfaces with boundary
TL;DR: In this paper, the authors studied the monoid of right-veering diffeomorphisms on a compact oriented surface with nonempty boundary, which strictly contains the monoids of products of positive Dehn twists and explained the relationship to tight contact structures and open book decompositions.
References
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Book
Topological methods in hydrodynamics
Vladimir I. Arnold,Boris Khesin +1 more
TL;DR: A group theoretical approach to hydrodynamics is proposed in this article, where the authors consider the hydrodynamic geometry of diffeomorphism groups and the principle of least action implies that the motion of a fluid is described by geodesics on the group in the right-invariant Riemannian metric given by the kinetic energy.
Book ChapterDOI
The asymptotic Hopf invariant and its applications
TL;DR: The classical Hopf invariant distinguishes among the homotopy classes of continuous mappings from the three-sphere to the twosphere and is equal to the linking number of the two curves that are the preimages of any two regular points of any regular point on the two-spheres.