F$ over the tangent of the oriented geodesic joining $p$ to $g(p)$. We claim that $\Psi(f)$ is a quasi-morphism for $\Gamma$.

For the proof of this claim, recall that the curvature of $\tilde M$ is pinched between two negative constants and therefore by comparison, for every $\epsilon_0>0$ there is a number $k=k(\epsilon_0)>0$ only depending on $\epsilon_0$ and an upper curvature bound for $\tilde M$ with the following property. Let $T$ be a geodesic triangle in $\tilde M$ with vertices $A_1,A_2,A_3$ and denote by $a_i$ the side of $T$ connecting $A_{i-1}$ to $A_{i+1}$. Let $q_i\in a_i$ be the nearest point projection of the vertex $A_{i}$ to the side $a_i$ and let $\gamma_{i,+},\gamma_{i,-}$ be the oriented geodesic arc parametrized by arc length which connects $q_i=\gamma_{i,+}(0)$ to $A_{i+1}=\gamma_{i,+}(\tau_{i,+})$, $q_i=\gamma_{i,-}(0)$ to $A_{i-1}=\gamma_{i,-}(\tau_{i,-})$ (here indices are taken modulo 3). Then $t_i=\tau_{i+1,+}-\tau_{i,-}\in [-k,k]$ and moreover for every $t\in [k,\tau_{i,-}]$ the distance between $\gamma_{i,-}^\prime(t)\in T^1\tilde M$ and $\gamma_{i+1,+}^\prime(t+t_i)\in T^1\tilde M$ is at most $\epsilon_0$.

Now by assumption, the function $F$ is anti-invariant under the flip and locally Hölder continuous. Therefore our above discussion implies that the integral of $\tilde
F$ over the unit tangent field of a closed curve in ${\rm Conv}(\Lambda)$ which consists of three geodesic 