of bounded negative curvature. Such a manifold $M$ contains a compact convex subset ${{\mathcal{C}}}(M)$, the so-called *convex core*, as a strong deformation retract. The group $\Gamma$ is a word hyperbolic, and the convex core ${{\mathcal{C}}}(M)$ of $M$ is a $K(\Gamma,1)$-space. Therefore, if $\Gamma$ is *non-elementary*, i.e. if $\Gamma$ is not abelian, then the dimension of the cohomology group $H^2(\Gamma,\mathbb{R})$ is finite, and by Fujiwara’s result [@F], the group $H_b^2(\Gamma,\mathbb{R})$ is infinite dimensional.

Inspired by a result of Barge and Ghys [@BG], we relate the second bounded cohomology group $H_b^2(\Gamma,\mathbb{R})$ to the *geodesic flow* $\Phi^t$ of $M$ which acts on the unit tangent bundle $T^1M$ of $M$. Since $M$ is convex cocompact, $T^1M$ admits a compact $\Phi^t$-invariant hyperbolic set $W$ which is the closure of the union of all closed orbits of $\Phi^t$. The restriction of $\Phi^t$ to $W$ is topologically transitive.

A *cocycle* for the action of $\Phi^t$ on $W$ is a continuous function $c:W\times \mathbb{R}\to \mathbb{R}$ such that $c(v,s+t)=c(v,t)+c(\Phi^tv,s)$ for all $v\in W$ and all $s,t\in
\mathbb{R}$. Two cocycles $c,d$ are *cohomologous* if there is a continuous function $\psi:W\to \mathbb{R}$ such that $\psi(\Phi^tv)+c(v,t)= d(v,t)+\psi(v)$. The collection of all cocycles which are cohomologous to a given cocycle $c$ is the 