Name of the modular group

Question

I've been studying the hyperbolic plane and the action of the group $PSL(2,\mathbb{R})$ on it. I found that the modular group $PSL(2,\mathbb{Z})$ is a discrete subgroup of $PSL(2,\mathbb{R})$ so it's Fuchsian.But where does the name of the Modular group come from?How is it related with the Modulii spaces?

Answer 1

If you wish to study flat structures on the torus $T^2 = \mathbb{R}^2 / \mathbb{Z}^2$, i.e. Euclidean metrics on $T^2$, it comes down to studying lattices in $\mathbb{R}^2$, i.e. discrete subgroups isomorphic to $\mathbb{Z}^2$.

Lattices can be normalized in some fashion. One useful normalization is to rotate and scale the lattice so that $(1,0)$ is a shortest element (not necessarily unique).

Suppose we have a lattice $L \subset \mathbb{R}^2$ has been normalized in that fashion, i.e. $e=(1,0)$ is a shortest element. Then the lattice is determined by its "modulus", which is the (almost) unique point $p=(x,y) \in L$ such that $|p| \ge 1$ (since $(1,0)$ is a shortest element) and $-\frac{1}{2} \le x \le \frac{1}{2}$ (since one can always add multiples of $(1,0)$.

Now if you work your way through these definitions you'll see that the set $$\{p = (x,y) \,\bigm|\, |p| \ge 1, \quad -\frac{1}{2} \le x \le \frac{1}{2} \} $$ turns out to be a fundamental domain for the action of $PSL(2,\mathbb{Z})$ on the upper half plane $\mathbb{H}^2$ by fractional linear transformation (and, of course, $PSL(2,\mathbb{R})$ is the whole group of orientation preserving isometries of $\mathbb{H}^2$). Furthermore, the non-uniqueness of the choice of $p$ (given the lattice) is exactly the gluing relation on the fundamental domain given by the two transformations under which the edges of this fundamental domain are identified with each other, those two transformations being $z \mapsto z+1$ and $z \mapsto -1/z$.

Thus, the quotient space $\mathbb{H}^2 / PSL(2,\mathbb{Z})$, also known as the "modular space", is the set of "moduli" of normalized flat structures on the torus, and the group $PSL(2,\mathbb{Z})$ is known as the "modular group".