Hyperbolic translation flow I am reading a paper which the author defines $\left\{ T_r \right\}$ as a hyperbolic translation along a geodesic $\gamma$ in the hyperbolic plane $\mathbb{H}^2$ with $\gamma(0) = 0$ such that $T_r (\gamma(0)) = \gamma(r) \ \  \forall r \in \mathbb{R}$.
I know that the hyperbolic translation depends on a curve and a point (I saw the definition of the hyperbolic translation in the book  "Introduction à la géométrie hyperbolique et aux surfaces de Riemann" on page 76). In this case, I see $T_r$ depends on the parameter $r$.
Would you help me understand this definition? How do I understand the  hyperbolic translation $T_{r_1} (x)$ for a $x \in \mathbb{H}^2$ and $r_1 \in \mathbb{R}$?
 A: Let us work in $U = \{ z \in \mathbb{C} : \textrm{Imag}(z) > 0 \}$: the upper half plane model of $\mathbb{H^2}$.  Note that $G = \textrm{SL}(2, \mathbb{R})$ acts on $U$ by Möbius transformations; the action is by isometries.  Consider the matrix
$$
T_\lambda = 
\pmatrix{
\lambda & 0 \\
    c   & 1/\lambda}
$$
where $\lambda$ is real and (say) greater than one.
[Note that here I am using slightly different notation that in the original question.  The "translation distance" of $T_\lambda$ is not equal to $\lambda$.]
It is an exercise to check that $T_\lambda$ is a hyperbolic element of $G$.  Also, its axis $I$ is the positive imaginary axis in $U$.  It is a further exercise to compute the translation distance of $T_\lambda$ restricted to $I$.
Let $L_\theta$ be the euclidean ray in $U$ that makes angle $\theta$ with $I$. It is an exercise to check that $T_\lambda$ also preserves $L_\theta$.  It is a further exercise to compute the translation distance of $T_\lambda$ restricted to $L_\theta$.  You should also check that $L_\theta$ is equidistant from $I$: that is, there is a fixed constant $d(\theta)$ so that every point $z \in L_\theta$ is distance $d(\theta)$ from some point $z' \in I$.
These equidistant lines $L_\theta$ answer your question.  It is a final exercise to transport the above discussion to a general hyperbolic element $T \in G$.
