More Hyperbolic Geometry

Today, I went to the first lecture with my REU advisor. I think he really likes number theory and told us a great deal about geometry, number theory, group theory and everything else. I have to say that there are insane things that occurs inside strange spaces that I cannot fathom even the tip of the iceberg because how ridiculous insane it is. Insane in a good way, of course.

First really surprising thing are the principal (principle?) congruence subgroups of $PSL(2, \mathbb{Z})$. These just have the strangest group structure. For one, $PSL(2, \mathbb{Z})$ has torsion element but some of these principal congruence subgroup (all of them?) are torsion free. The cooler thing is that whether the subgroup is torsion free or not relates to something about the fundamental domain of the subgroup. $PSL(2, \mathbb{Z})$ contains 2 elliptic point, namely, $i$ and $(1/2, \sqrt{3}/2)$. These two points somehow corresponds to $PSL(2, \mathbb{Z})$‘s torsion element of order 2 and 3.

Equally surprising are the fundamental domains of these principal congruence subgroups, and their quotient spaces. Much like in the Euclidean plane, in the hyperbolic plane, we can form these quotient spaces. Whereas $\mathbb{Z}^2 \backslash \mathbb{R}^2$ is a torus, $\Gamma(N) \backslash \mathbb{H}^2$ are these punctured spheres and toruses of genus 2, 3 or beyond.

Another thing that I can not yet grasp is the different isometric classes of toruses. A torus made from a square lattice has less symmetry than a hexagonal lattice torus. Similarly, there are quotient spaces in $\mathbb{H}^2$ that have a lot more symmetry than the others. The meaning of this is something I will explore in the next month or so.

Finally, what is the curvature on a torus? In fact, what is curvature? I have no studied this property yet, but I am seeing it everywhere. Gauss discovered that curvature is intrinsic to the object. Yet, is torus flat in $\mathbb{R}^4$ but has no imbedding into $\mathbb{R}^3$ that is flat? The meaning of this I will explore in the next month or so.