Invariant lifts of a closed curve on a surface of genus > 1 I am learning some things about surfaces of genus greater than $1$, and I am trying to answer this question :
Let $S$ be a compact and orientable surface of genus $g \geq 2$, and $c$ a closed curve on $S$. Let $\widehat{S}$ denote the universal covering space of $S$. Given a deck transformation $\widehat{f}: \widehat{S} \rightarrow \widehat{S}$, is it true that there exists only a finite number of lifts $\widehat{c}$ of $c$ to $\widehat{S}$ that are $\widehat{f}-$invariant ?
(by "lift", I mean the image of a  map $\widehat{c} : \mathbb{R} \rightarrow \widehat{S}$ that lifts the application $c : \mathbb{R}/\mathbb{Z} \rightarrow S$ defining the closed curve $c$).
I hope this is clear ! Thank you for any help :) 
 A: Yes. If you pick a basepoint $b \in c$ then your curve represents an element of the fundamental group $\pi_1(S,b)$, and therefore also gives rise to a corresponding deck transformation $\hat f$. In particular, if you pick a basepoint $\hat b$ of the universal cover $\hat S$, then this deck transformation will map the lift $\hat c_{\hat b} \ni \hat b$ of $c$ to itself.
Conversely for any deck transformation $\hat f$ you can find a curve $c$ that realizes it in $\pi_1$.
Now some hyperbolic geometry is needed. You can endow $S$ with a hyperbolic metric (which lifts to the universal cover) and assume that $c$ was geodesic, that is to say that for any lift $\hat c$ of $c$, for any points $x,y$, the arc on $\hat c$ connecting $x,y$ is the shortest path between them. Unlike in Euclidean space where parallel lines stay close, hyperbolic geometry implies that no two distinct infinite geodesics can remain a bounded from one another. 
Because the images of all lifts of $c$ correspond to distinct geodesics, there is only one of these geodesics that will be fixed by your deck transformation. This is viewing geodesics as subsets of the universal cover. Lifts, correspond to functions of $\mathbb R$ and it is possible to get multiple lifts being fixed by a deck transformation if $c$ is a power of a curve, but this number is exactly the power.
