I am trying to understand better Geodesics in $X:= \mathbb{H}/\mathrm{SL}_2(\mathbb{Z})$, where $\mathbb{H}$ is the hyperbolic plane. I am using the upper half complex plane as a model.
In this setting, the geodesics are half circles, with center in $\mathbb{R}$, or the lines perpendicular to the real axis.
The question is about the way these half circles (or lines) in $\mathbb{H}$ behave projected in $X$.
Their projection can either be of finite length or infinite and this is what I am trying to caracterize. The elements of $\mathrm{SL}_2(\mathbb{Z})$ act on $\mathbb{H}$ via Möbius transformations of the form:
$$\tau \in \mathbb{H} \overset{A}{\mapsto} \frac{a \tau + b}{c \tau + d} \in \mathbb{H} \qquad \text{for any } A = \begin{pmatrix} a & b \\ c & d \end{pmatrix} \in \mathrm{SL}_2(\mathbb{Z})$$
These are isometries and therefore map geodesics to geodesics.
Hyperbolic elements of $\mathrm{SL}_2(\mathbb{Z})$ are exactly those with two Fixed points lying in $\mathbb{R} \cup \{ \infty \}$. They are also exactly caracterized by the fact that $ |\mathrm{tr}(A) | >2$. These apparently play a very important role in caracterizing the geodesics that have finite length, which is I am trying to figure out.
My take goes as follows:
If a half circle $C \subset \mathbb{H}$ gets projected into a geodesic of finite length,then, starting at some point $p$ on $C$ and moving along $C$ with unit speed (call that curve $c: [0,\infty) \rightarrow C$, there must be a point $L$ at which the projection into a Fundamental domain $\Gamma$ satisfies $\pi(p) = \pi(c(0)) = \pi(c(L))$ and $\pi(p)' = \pi(c(L))'$. And if we continue moving along $C$, we just keep moving in $\pi(c([0,L]))$.
We can choose such an $L$ minimal and we can conclude from $\pi(c(0)) = \pi(c(L))$ that there is an element $\gamma \in \mathrm{SL}_2(\mathbb{Z})$ such that $ p =c(0) \overset{\gamma}{\mapsto} c(L)$ and by our construction, this is true for any $p \in C$, so it seems that such an element maps $C$ into itself and $\gamma$ pushes $L$ into the future.
Another property of $\mathrm{SL}_2(\mathbb{Z})$ is that they map $\mathbb{R}$ into itself. Therefore, if we call $z,w \in \mathbb{R}$ the two points extremities of $C$, this property of $\gamma$ can either permute $z$ and $w$ or fix them both. If $\gamma$ fixes them, then $\gamma$ is hyperbolic (because its two Fix points are real).
If it permutes $z$ and $w$, then by the intermediate value theorem, $\gamma$ must have some Fixed point in $a \in C$ ($a \overset{\gamma}{\mapsto} a$). This would however mean that for a path $c(t)$ like before starting at $a$, we would have $c(0) = c(L)$, a contradiction. Therefore, we conclude that $\gamma$ must be hyperbolic with $z,w$ as Fix points.
Is there something wrong with my reasoning?
(I think the converse than also be shown, which I might write down later on)
P.S. This question is a refinement of this Post.