anti-invariant function on $T^1M$.*]{}

[*Proof:*]{} Let $f\in {{\mathcal{A}}}$ and let $F$ be a locally Hölder continuous flip anti-invariant extension of $f$ to $T^1M$. Denote by $\Psi(f)$ the quasi-morphism constructed in the proof of Lemma 2.1 using the extension $F$ of $f$ and the basepoint $p\in {\rm Conv}(\Lambda)$. We first show that a different choice $q\in {\rm Conv}(\Lambda)$ of a basepoint gives rise to a quasi-morphism which is equivalent to $\Psi(f)$ in our above sense. For this we follow [@BG]. Let $\tilde F$ be the lift of $F$ to $T^1\tilde M$. For $g\in \Gamma$ define $\rho(g)$ to be the integral of the function $\tilde F$ over the unit tangent field of the oriented geodesic quadrangle with vertices $q,g(q),g(p),p$. As in the proof of Lemma 2.1 we conclude that the function $\rho:\Gamma\to \mathbb{R}$ is uniformly bounded. Since by invariance of $\tilde F$ under the action of $\Gamma$ the integral of $\tilde F$ over the oriented geodesic arc connecting $g(q)$ to $g(p)$ is independent of $g\in
\Gamma$ and, in particular, it coincides with the negative of the integral of $\tilde F$ over the oriented geodesic arc connecting $p$ to $q$, the quasi-morphism defined by $F$ and the basepoint $q$ just equals $\Psi(f)+\rho$. Thus 