Transitivity of group action of surface group on the geodesic in universal cover

Question

Consider a closed hyperbolic surface of genus $n$, with a geodesic at least immersed on the surface- i.e. intersections are allowed. Consider the lift of this geodesic to the universal cover of the surface. Is the action of the fundamental group on the lift of geodesic in the universal cover transitive?

Answer 1

Yes, the deck transformation group acts transitively on the lifts of the geodesic.

For the proof, take a point $p$ on the geodesic $\gamma$ which is not an a self-intersection point.

Take the entire lift of that point to the universal cover, call that set $P$. The deck transformation group acts transitively on $P$ (that's one reason why we picked $p$ to not be a self-intersection point).

For any two lifted geodesics $\tilde\gamma_1,\tilde\gamma_2$, pick $p_1,p_2 \in P$ so that $p_i \in \tilde\gamma_i$. Choose a deck transformation $g$ such that $g\cdot p_1 = p_2$. Thus, $g \cdot \tilde\gamma_1$ and $\tilde\gamma_2$ are each lifts passing through $p_2$. But there is a unique lift of $\gamma$ through $p_2$ (that's another reason why we picked $p$ to not be a self-intersection point), and so $g \cdot \tilde\gamma_1 = \tilde\gamma_2$.