the quasi-morphism $\Psi(f)$. By Lemma 2.2, this class only depends on $f$. Moreover, the assignment $\Theta:{{\mathcal{A}}}\to \tilde Q$ is clearly linear. We next investigate the kernel of the map $\Theta$.

Since $\Gamma$ is convex cocompact by assumption, there is a natural correspondence between oriented closed geodesics on $M$ and conjugacy classes in $\Gamma$. For every homomorphism $\rho:\Gamma\to \mathbb{R}$ and every $g\in \Gamma$, the value $\rho(g)$ of $\rho$ on $g$ only depends on the conjugacy class of $g$. Therefore such a homomorphism defines a function on the set of closed geodesics on $M$; we denote the value of $\rho$ on such a closed geodesic $\gamma$ by $\rho(\gamma)$. We have.

[**Lemma 2.3:**]{} [*$\Theta(f)=0$ if and only if there is a morphism $\rho:\Gamma\to \mathbb{R}$ such that $\int_{\gamma^\prime} f =\rho(\gamma)$ for every closed geodesic $\gamma$ on $M$.*]{}

[*Proof:*]{} Let $f\in {{\mathcal{A}}}$ and assume that there is a morphism $\rho:\Gamma\to \mathbb{R}$ such that $\int_{\gamma^\prime} f=\rho(\gamma)$ for every closed geodesic $\gamma$ on $M$. This morphism defines a class in $H^1(M,\mathbb{R})$ and therefore by the de Rham theorem, there is a smooth closed 1-form $\beta$ on $M$ which defines $\rho$ via integration. Let $\tilde \beta$ be the pull-back of $\beta$ to a closed $1$-form on $\tilde M$. 