infinite dimensional second bounded cohomology for a countable group $\Gamma$ acting as a group of homeomorphisms on a metric space of bounded diameter. The coefficients of these cohomology groups can be either $\mathbb{R}$ or $\ell^p(\Gamma)$ for some $p\in (1,\infty)$. Theorem 4.4 of Section 4 shows that our criterion can be applied to countable groups which admit a non-elementary weakly acylindrical isometric action on a hyperbolic geodesic metric space; this then yields Theorem A. Section 5 contains the proof of Corollary B and Corollary C as well as a short discussion of some applications which are due to Monod and Shalom.

Dynamical cocycles and bounded cohomology
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In this section we consider an $n$-dimensional convex cocompact Riemannian manifold $M$ of bounded negative curvature. Then $M=\tilde M/\Gamma$ where $\tilde M$ is a simply connected complete Riemannian manifold of bounded negative curvature and $\Gamma$ is a group of isometries acting properly discontinuously and freely on $\tilde M$. The manifold $\tilde M$ admits a natural compactification by adding the *geometric boundary* $\partial \tilde M$ which is a topological sphere of dimension $n-1$. Every isometry of $\tilde M$ acts on $\partial \tilde M$ as a homeomorphism. The *limit set* $\Lambda$ of $\Gamma$ is the set 