Is it true that a geodesic on a hyperbolic surface can be lifted to a geodesic on the hyperbolic plane?

Question

Let $\mathbb{H}$ be the hyperbolic plane, $\Gamma < \text{Isom}(\mathbb{H})$ be a Fuchsian group, and $S = \mathbb{H}/\Gamma$. If $\gamma : [0,1] \rightarrow S$ is a geodesic, can it be lifted to a geodesic $\tilde{\gamma} : [0,1] \rightarrow \mathbb{H}$?

Answer 1

Yes!

Denote by $\pi$ the covering map $\pi: \mathbb{H} \to S$.

There are many lifts of $\gamma$ corresponding to the many choices of the starting point $\tilde{\gamma}(0) \in \pi^{-1}(\gamma(0))$. But after one chooses a starting point, there is a unique lift of $\gamma$ to a curve $\tilde{\gamma}$ since $\pi$ is a covering map (this follows from a general fact in topology; see, e.g., Wikipedia article on covering spaces).

Furthermore, since $\pi$ is a local isometry and the property of being a geodesic is a local property, $\tilde{\gamma}$ is also a geodesic.