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The translation number and quasi-morphisms on groups of symplectomorphisms of the disk. (Le nombre de translation et les quasi-morphismes sur des groupes de symplectomorphismes du disque.) (English. French summary) Zbl 1522.53065

This paper is concerned with the construction of homogenous quasi-morphisms on groups of symplectomorphisms of the disk.
Let \(D=\{ (x,y) \in \mathbb R^2\mid x^2 + y^2 \leq 1 \}\) be the unit disk in \(\mathbb R^2\) and \(\omega = dx \wedge dy \) be the standard symplectic form on \(D\). Let \(G= \mathrm{Symp}(D)\) be the group of symplectomorphisms of \(D\) (which may not be the identity on the boundary \(\partial D\)). In this paper the author first constructs a homogeneous quasi-morphism on \(G\), extending the Calabi invariant. Recall that the restriction homomorphism \(\rho: G \to \mathrm{Diff}_+ (S^1)\), where \(\mathrm{Diff}_+ (S^1)\) is the group of orientation-preserving diffeomorphisms of the unit circle \(S^1 = \partial D\), is surjective, see [T. Tsuboi, Trans. Am. Math. Soc. 352, No. 2, 515–524 (2000; Zbl 0937.57023)]. Denote by \(G_{\mathrm{rel}}\) the kernel of \(\rho\). The Calabi invariant \(\mathrm{Cal}: G_{\mathrm{rel}} \to \mathbb R\) is defined by \[ \mathrm{Cal}(h) = \int_D h^*\eta \wedge \eta, \] where \(\eta\) is a \(1\)-form satisfying \(d\eta = \omega\). It is well known that this invariant is a surjective homomorphism and it is independent of the choice of \(\eta\). The author defines the map \(\tau_\eta: G \to \mathbb R\) in the same way: \[ \tau_\eta (g) = \int_D g^*\eta \wedge \eta. \] This map is not a homomorphism and does depend on \(\eta\). It turns out that it is a quasi-morphism, its homogenization \(\bar{\tau}\) does not depend on \(\eta\) and it is an extension of the Calabi invariant. Moreover, there is another extension of the Calabi invariant previously introduced by T. Tsuboi [loc. cit.], which is a homomorphism to \(\mathbb R\) from the universal covering group of \(G\). One of the main results of this paper (Theorem 1.1) establishes a relation between the two extensions, involving the translation number introduced by H. Poincaré [C. R. Acad. Sci., Paris 90, 673–675 (1880; JFM 12.0588.01)].
In a similar way, the author constructs a homogeneous quasi-morphism \(\bar{\sigma}\) on the subgroup \(G_o\) of \(G\), consisting of symplectomorphisms preserving the origin, which extends a version of the flux homomorphism. More precisely, let \(G_{o,\mathrm{rel}} = G_{\mathrm{rel}} \cap G_o\). The flux homomorphism \(\mathrm{Flux}_{\mathbb R}\) is defined by \[ \mathrm{Flux}_{\mathbb R}(h) = \int_\gamma h^*\eta - \eta, \] where \(\gamma\) is a path from the origin \(o\) to a point in the boundary \(\partial D\). This homomorphism is surjective and it is independent of the choice of \(\eta\) and \(\gamma\). As in the previous case, the quasi-morphism \(\bar{\sigma}\) relates with another extension of the flux homomorphism through the translation number (Theorem 1.2).
Finally, in the last section, the author shows that the difference \(\bar{\tau} -\pi \bar{\sigma}: G_{o} \to \mathbb R\) is a continuous homomorphism, extending the difference \(\mathrm{Cal} -\pi \mathrm{Flux}_{\mathbb R}\), although \(\mathrm{Cal}\) and \(\mathrm{Flux}_{\mathbb R}\) cannot can be extended to homomorphisms on \(G_o\).

MSC:

53D05 Symplectic manifolds (general theory)
58D05 Groups of diffeomorphisms and homeomorphisms as manifolds
20J06 Cohomology of groups
53D22 Canonical transformations in symplectic and contact geometry
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