The paper under review is devoted to the study of the group of Hamiltonian diffeomorphisms of a symplectic manifold \((M, \omega)\). Under some hypothesis, the authors define a non-trivial cocycle on this group with values in \({\mathcal C}^{\infty}(M)/{\mathbb R}\) and use it to get informations about its finitely generated subgroups. More precisely, if \((M,\omega)\) is exact and convex at infinity and \(\Gamma\) is a finitely generated subgroup of \(\text{Ham}_{c}(M,\omega)\) whose action on \(M\) is effective then any non-trivial cyclic subgroup of \(\Gamma\) is undistorted (in the sense of [M. Gromov, Geometric group theory. Volume 2: Asymptotic invariants of infinite groups. London Mathematical Society Lecture Note Series. 182. Cambridge: Cambridge University Press (1993; Zbl 0841.20039)]). By passing to the universal cover of \(M\), Kȩdra and Gal obtain the same result for closed symplectic manifolds \((M,\omega)\) which are symplectically hyperbolic, i.e. symplectically aspherical (\(\omega|_{\pi_{2}(M)}=0\)) and such that a primitive of \(\omega\) on the universal cover of \(M\) is bounded with respect to the pullback of a Riemannian metric on \(M\). This result was also proved by L. Polterovich [Invent. Math. 150, No. 3, 655–686 (2002; Zbl 1036.53064)].