### A bit of computations

Today, I did a pretty nifty computation to explore the behavior of $f(z) = \frac{z}{z+1}$. This transformation is a linear fractional transformation on the upper half plane with fixed point 0. The blue circle is the horocycle on which the points $\{f^n (i) | n \in \mathbb{Z} \}$ lies. The yellow semi-circle is the unit circle. The region bounded by the vertical red line, vertical blue line and the exterior of the yellow semi-circle, is the modular figure, which is the fundamental domain of $\Gamma = PSL(2, \mathbb{Z})$.

Some questions I have about the group $\Gamma$ is regarding the parabolic transformations: how many degrees of freedoms do we have in the “lattice”? In other word, since the parabolic transformations are the hyperbolic analog of translations, there should be a set of directions to travel in. Of course, none of the directions should cross path. Besides parabolic transformations, the hyperbolic transformations are also not torsion. Hence, the term “direction” should be a bit more complicated here.

Actually, does it even make sense to speak of a hyperbolic lattice? In the Euclidean plane, the translation subgroup of a wallpaper group can generate a lattice by translating the origin. Similarly, we would like to study the lattice generated by translating $i$. I believe this question will illuminate the structures of the congruence subgroups of $\Gamma$.