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?
The answer to your question depends on the behaviour of $\gamma$ in $M$. The following claims are easy to check.
There exists a deck transformation $\alpha$ such that $\alpha\tilde\gamma$ is a reparametrisation of $\tilde\gamma$ if and only if $\gamma$ is closed.
There exists $\alpha$ such that $\tilde\gamma$ intersects $\alpha\tilde\gamma$ if and only if $\gamma$ has a selfintersection.
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).
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.