changing the basepoint does not change the equivalence class of our quasi-morphism $\Psi(f)$.

Now we may also replace the point $q\in {\rm Conv}(\Lambda)$ by a point $\xi\in \Lambda$. Namely, for $g\in \Gamma$ define $\rho(g)$ to be the oriented integral of the function $\tilde F$ over the tangent lines of the ideal geodesic quadrangle with vertices $\xi,g(\xi),g(p),p$. As before, this function is uniformly bounded. By invariance of $\tilde F$ under the action of $\Gamma$ and the fact that $\tilde F$ is anti-invariant under the flip we obtain that the 2-cocycle for $\Gamma$ defined as above by the quasi-morphism $\Psi(f)+\rho$ is just the cocycle $\eta\in
L^\infty(\Gamma^3,\mathbb{R})^\Gamma$ which assigns to a triple $(g,h,u)\in \Gamma^3$ the integral of $\tilde F$ over the unit tangents of the (possibly degenerate) oriented ideal triangle with vertices $g(\xi), h(\xi),u(\xi)$. Since these unit tangents are contained in the lift $\tilde L$ of the non-wandering set $L$ for the geodesic flow on $T^1M$, the cocycle $\eta$ only depends on $f$ but not on an extension of $f$ to $T^1M$. Thus the cohomology class defined by $\Psi(f)$ is independent of the extension as well.

In the sequel we denote for $f\in {{\mathcal{A}}}$ by $\Theta(f)\in \tilde{{\mathcal{Q}}}\sim{\rm
ker}(H_b^2(\Gamma,\mathbb{R})\to H^2(\Gamma,\mathbb{R}))$ the cohomology class of 