associates to a quasi-morphism $\phi$ its *defect* $\Vert \phi\Vert_0$ is a pseudo-norm which vanishes precisely on the subspace of *morphisms*.

[**Lemma 2.1:**]{} [*There is a linear map $\Psi:{{\mathcal{A}}}\to
{{\mathcal{Q}}}$. For every $f\in {{\mathcal{A}}}$, the defect $\Vert \Psi(f)\Vert_0$ of $\Psi(f)$ is bounded from above by a constant only depending on the curvature bounds of $M$ and the Hölder norm of $f$.*]{}

[*Proof:*]{} Let $f\in {{\mathcal{A}}}$ and extend $f$ to a locally Hölder continuous flip anti-invariant function $F$ on $T^1M$. Such an extension can be constructed as follows. Choose a probability measure $\mu$ on $L$ for which there are constants $0<a<b$ such that the $\mu$-mass of a ball $B(v,r)$ of radius $r<1$ about a point $v\in L$ is contained in $[r^b,r^a]$; for example, the unique measure of maximal entropy for the geodesic flow on $L$ has this property. We view $\mu$ as a probability measure on $T^1M$ which is supported in $L$. Let $\tau:[0,\infty)\to
[0,1]$ be a smooth function which satisfies $\tau(t)=1$ for $t$ close to $0$ and $\tau[1,\infty)=0$. Via multiplying the restriction of $\mu$ to the metric ball $B(w,r)$ $(w\in T^1M)$ by the function $z\to \tau(d(z,w)/r)$ we may assume that the measures $\mu\vert B(w,r)$ depend continuously on $w\in T^1M,r>0$ in the 