I am trying to solve an Exercise in the book Audrey Terras - Harmonic Analysis on Symmetric Spaces and Applications I-Springer (1985) and I struggle to find good justifications.
In the exercise, we consider the hyperbolic plane (model is the upper half plane $(x,y), y > 0$, with metric $g_{ij} = \delta_{ij}/y^2$) and $C(z,w)$ is a geodesic path (meaning a half circle or straight line perpendicular to the "x-axis") joining $z,w \in \mathbb{R} \cup \{\infty \}$. $\overline{C}(z,w)$ is the projection into $\mathbb{H}/\mathrm{SL}_2(\mathbb{Z})$ of said path and closed is defined (in the exercise) through the existence of a parametrization (say $\overline{c}$ in $X:= \mathbb{H}/\mathrm{SL}_2(\mathbb{Z})$ of $\overline{C}(z,w)$ such that $\overline{c}(t)$ has the same begining and end points.
The first question of the problem has two parts, I must show that:
- $\overline{C(z,w)}$ is closed $\iff$ there is $\gamma \in \mathrm{SL}_2(\mathbb{Z})$ such that $\gamma C(z,w) \subset C(z,w)$.
- $\overline{C(z,w)}$ is closed $\iff$ $z,w$ are fixed points of a hyperbolic element of $\gamma$ of $\mathrm{SL}_2(\mathbb{Z})$.
When discussing the Problem, I was told that it is an issue if we go through vertices of a Fundamental domain, or some Fix Point of an element of $\mathrm{SL}_2(\mathbb{Z})$, but I don't get it and I haven't found good references to get a better understanding of this specific issue, so that would be welcome.
The main analogy I work with is that of Lattices (in $\mathbb{R}\times \mathbb{R}_{>0}$, euclidian). Geodesics are any lines ($y= ax+b$) and projecting them into a fundamental domain (a square with opposite sides identified / a torus) accordingly, but this doesn't seem to reflect this ambiguity...
Here goes what was my attempt at a proof of one implication:
- $"\implies"$: $C(z,w)$ is infinitely long, but it's projection has finite length. Therefore, if we call $[\cdot]$ the projection into a fundamental domain, at any point $y \in C(z,w)$, we can find $d \in \mathbb{R}_{>0}$ such that for an arc length path $c : [0,d]$ in $C(z,w)$ starting at $y$, the map \begin{align*} \overline{c_y} : [0,d] &\rightarrow X, \\ t &\mapsto [c(s+t)] \end{align*} isn't injective and we can choose $d$ to be minimal. For $L$ such a minimum, it follows that $[c_y]$ is $L$-periodic in $\mathbb{H}/\mathrm{SL}_2(\mathbb{Z})$ for any starting point. Therefore, $[c_y(0)] = [c_y(L)]$ and $c_y(0) \neq c_y(L)$. By definition, this means that there exists $\gamma \in \mathrm{SL}_2(\mathbb{Z})$ such that $c_y(0) = \gamma c_y(L)$ for all $y \in C(z,w) \implies \gamma C(z,w) \subset C(z,w)$.
P.S. I may write down my attemps for all of 1. and 2. at some point
Here is the Problem (Ex. 20, p277) as found in the book (don't mind the highlights) :
