hence the set ${{\mathcal{Q}}}B$ of equivalence classes of quasi-morphisms of $\Gamma$ has a natural structure of a vector space. It contains as a subspace the vector space $H^1(\Gamma,\mathbb{R})$ of all equivalence classes of *morphisms* of $\Gamma$. There is an exact sequence $$0\to H^1(\Gamma,\mathbb{R})\to {{\mathcal{Q}}}B\to
H_b^2(\Gamma,\mathbb{R}) \to H^2(\Gamma,\mathbb{R})$$ and therefore the quotient space $\tilde {{\mathcal{Q}}}={{\mathcal{Q}}}B
/H^1(\Gamma,\mathbb{R})$ can naturally be identified with the kernel of the map $H_b^2(\Gamma,\mathbb{R})\to
H^2(\Gamma,\mathbb{R})$ (see [@M]). In particular, an equivalence class of quasi-morphisms can be viewed as a cohomology class of $\Gamma$-invariant bounded cocycles $\phi\in
L^\infty(\Gamma^3,\mathbb{R})^\Gamma$. In this interpretation, the cocycle $\phi$ determined by the quasi-morphism $\psi$ associates to a triple $(g,h,u)\in \Gamma^3$ the value $\phi(g,h,u)=\psi(g^{-1}h)+\psi(h^{-1}u)-\psi(g^{-1}u)$.

For $f\in {{\mathcal{A}}}$, the definition of the quasi-morphism $\Psi(f)$ in the proof of Lemma 2.1 depends on the choice of an extension of $f$ to a locally Hölder continuous flip anti-invariant function on $T^1M$ and also on the choice of a basepoint $p\in {\rm Conv}(\Lambda)$. The next lemma shows that the cohomology class of $\Psi(f)$ is independent of these choices.

[**Lemma 2.2:**]{} [*The cohomology class of the quasi-morphism $\Psi(f)$ does not depend on the choice of the basepoint $p$ nor on the extension of $f$ to a locally Hölder continuous flip 