I have a question regrading the deck transformations. As we know, for the 2-torus $\mathbb T^2$, if we have a geodesic $\widetilde\gamma$ on the corresponding covering space $\mathbb R^2$, and $\alpha$ is a deck transformation not equal to the identity, then $\widetilde\gamma$ and $\alpha(\widetilde\gamma)$ are parallel and separated by a uniform constant(1 in this case).
I am wondering if we can get some similar results for the hyperbolic surfaces. Basically I am asking if we have a hyperbolic surface $M$ with covering space $\mathbb H^2$, $\alpha$ is a non-identity deck transformation, do we have some result like what happens on the 2-torus? Can $\widetilde\gamma$ and $\alpha(\widetilde\gamma)$ intersects each other? Can $\widetilde\gamma$ and $\alpha(\widetilde\gamma)$ be parallel but becoming closer and closer at infinity? Are they alywas separated by some uniform constant?



Geodesics under deck transformations on hyperbolic surface
The answer to your question depends on the behaviour of $\gamma$ in $M$. The following claims are easy to check.


1. There exists a deck transformation $\alpha$ such that $\alpha\tilde\gamma$ is a reparametrisation of $\tilde\gamma$ if and only if $\gamma$ is closed.
2. There exists $\alpha$ such that $\tilde\gamma$ intersects $\alpha\tilde\gamma$ if and only if $\gamma$ has a selfintersection.
3. There exists $\alpha$ such that $\tilde\gamma$ and $\alpha\tilde\gamma$ are asymptotic if either $\gamma$ is asymptotic to a closed geodesic (then $\alpha$ is hyperbolic) or if $\gamma$ runs into a cusp (then $\alpha$ is parabolic).
4. If $\gamma$ is simply closed, then each decktransformation $\alpha$ either acts as in 1., or $\alpha\tilde\gamma$ and $\tilde\gamma$ are separated by a uniform distance.


