Bi-invariant metrics and quasi-morphisms on groups of Hamiltonian diffeomorphisms of surfaces

Abstract

Abstract. Let Σ g be a closed orientable surface of genus g and let Diff 0 (Σ g , area) be the identity component of the group of areapreserving diffeomorphisms of Σ g . In this work we present the extension of Gambaudo-Ghys construction to the case of a closed hyperbolic surface Σ g , i.e. we show that every non-trivial homogeneous quasi-morphism on the braid group on n strings of Σ g defines a non-trivial homogeneous quasi-morphism on the group Diff 0 (Σ g , area). As a consequence we give another proof of the fact that the space of homogeneous quasi-morphisms on Diff 0 (Σ g , area) is infinite dimensional.Let Ham(Σ g ) be the group of Hamiltonian diffeomorphisms of Σ g . As an application of the above construction we construct two injective homomorphisms Z m → Ham(Σ g ), which are bi-Lipschitz with respect to the word metric on Z m and the autonomous and fragmentation metrics on Ham(Σ g ). In addition, we construct a new infinite family of Calabi quasi-morphisms on Ham(Σ g ).

Fragmentation norm and relative quasimorphisms

“…The proof of the above claim is straightforward and relies on the observation that the subsets of the supports of the push maps where they vary between the identity and the rotation can be made arbitrarily small. It is similar to the proof of Lemma 3.1 and similar arguments are presented in [2,14].…”

On the autonomous norm on the group of Hamiltonian diffeomorphisms of the torus

“…Historically the braid approach was the first original idea due to Gambaudo and Ghys [8] applied to diffeomorphisms of the disc and the sphere. It was later generalized by the first named author to other surfaces [2]. To sum up, the construction gives a linear map G : Q(P n (Σ)) → Q(Ham(Σ)), from the space of homogeneous quasimorphisms on the pure braid group to the space of homogeneous quasimorphisms on the group of Hamiltonian diffeomorphisms of the surface.…”

Entropy and quasimorphisms

“…Embedding Theorem. The proof of the following theorem is a variation of the proof of Ishida [19], see also [8,10]. We present it for the reader convenience.…”

“…In the first part of the paper we revise and extend the construction of quasimorphisms on Diff 0 (S, area) given by Gambaudo-Ghys [17] and Polterovich [23], see also [7,8]. The main advantage of our approach is that it allows to treat all surfaces in an unified way and to show there are infinitely many linearly independent homogeneous quasimorphisms on Diff(S, area) whose restrictions on Diff 0 (S, area) are linearly independent.…”

“…There exists another conjugation invariant word norm on Ham(S), the autonomous norm. It is unbounded in the case when S is a compact oriented surface, see [8,9,10,11,17]. Theorem 2 together with the fact that every autonomous diffeomorphism of a surface has entropy zero, see [29], gives a new proof of unboundedness of the autonomous norm on Ham(S).…”