arcs forming a geodesic triangle is bounded from above in absolute value by a universal constant times the Hölder norm of the restriction of $F$ to the compact subset $T^1M\vert {{\mathcal{C}}}(M)$ of $T^1M$ of all unit vectors with foot point in the convex core ${{\mathcal{C}}}(M)={\rm Conv}(\Lambda)/\Gamma$. On the other hand, by invariance of $\tilde F$ under the action of $\Gamma$ and by anti-invariance of $\tilde F$ under the flip, for $g,h\in \Gamma$ the quantity $\vert \Psi(f)(g)+\Psi(f)(h)-\Psi(f)(gh)\vert$ is just the absolute value of the integral of $\tilde F$ over the unit tangent field of the oriented geodesic triangle in ${\rm Conv}(\Lambda)\subset \tilde M$ with vertices $p,g(p),g(h (p))$. Thus $\Psi(f)$ is indeed a quasi-morphism and the assignment $f\to
\Psi(f)$ defines a linear map $\Psi:{{\mathcal{A}}}\to {{\mathcal{Q}}}$. Moreover, the defect $\Vert \phi\Vert_0$ of $\phi$ is bounded from above by a constant only depending on the curvature bounds of $M$ and the Hölder norm of $f$. This shows the lemma.

Two quasi-morphisms $\phi,\psi$ for $\Gamma$ are called *equivalent* if $\phi-\psi$ is a bounded function. This is clearly an equivalence relation. If $\phi_1$ is equivalent to $\phi_2$ and $\psi_1$ is equivalent to $\psi_2$ then for all $a,b\in \mathbb{R}$ the quasi-morphism $a\phi_1+b\psi_1$ is equivalent to $a\phi_2+b\psi_2$ and 