What is the quotient space $\mathbb{H}/PSL(2,\mathbb{Z})$?

Question

I was wondering how to describe this quotient(it's genus, marked points etc.). I think that this quotient is just a infinite strip that goes along the vertical axis. Even if so I dont know what to say about it structure, i can propose that it has genus 0...

Answer 1

Exercise 4 re-numbered:

(i) Show that each point of $ℍ/PSL(2, ℤ)$ has a representative inside the “strip”$$\{τ ∈ ℍ : |Re(τ)| ≤ 1/2, |τ| ≥ 1\}$$and then checking that the only remaining identifications are on the boundary of the strip. (ii) Show that $ℍ / PSL(2, ℤ)$ is a topological space homeomorphic to $ℂ$.

(i) is Thm 1.1 in an expository paper by Keith Conrad

My attempt to prove (ii) :

From (i) we know the remaining identifications are $(\frac12,y),(-\frac12,y)$ for $y\ge1$ and $(\cos\theta,\sin\theta),(-\cos\theta,\sin\theta)$ for $\theta\in[\pi/3,2\pi/3]$.

Folding the strip horizontally, we get a one-sided infinite cone which is homeomorphic to $\mathbb C$.

Answer 2

You can see as hbghlyj pointed out that your quotient is homeomorphic to the finite plane $\mathbb{C}$. An argument has already been given in a somewhat intuitive manner, as the strip-like set of hbghlyj's answer acts as a fundamental domain for the action of $PGL(2,\mathbb{Z})$. If we denote it by $F$ then $$F/PGL(2,\mathbb{Z})\simeq \mathbb{H}/PGL(2,\mathbb{Z})$$ and we see that the left hand side quantity is a one-sided infinite cone, by identifying the sides of the boundary of $F$ in a proper manner.

If you wish to see a more robust proof though, you may check out the book ``Complex Functions, an algebraic and geometric viewpoint'', page 285, Theorem 6.5.5. for a demonstration using the J-invariant of Klein.

I hope this is helpful!