$\Gamma(1)\backslash \Bbb H$ quotient meaning

Question

Let $\Bbb H$ be the upper-half complex plane and $\Gamma(1)=SL_2(\mathbb{Z})$ (acting on $\Bbb H$ by Möbius tranformations) . My modular forms notes frequently refer to the quotient $\Gamma(1) \backslash \Bbb H$, but I'm not quite sure what this quotient means, since the upper-half plane is not a subgroup (at least on first glance). Is it maybe related to orbits, or to a fundamental set? Also, what's the meaning of the backslash? Is it some notation that I'm not aware of?

Answer 1

There is a group action of $\Gamma(1)$ on $\mathbb{H}$ given by $\begin{bmatrix} a& b \\ c&d \end{bmatrix}z=\frac{az+b}{cz+d}$. To answer your question on the slash, this is a left group action, so when we write $\Gamma(1)\backslash\mathbb{H}$ we are taking equivalence classes of $\mathbb{H}$ modulo the action of $\Gamma(1)$. For example, we have that $$T=\begin{bmatrix}1&1\\0&1\end{bmatrix}$$ is a matrix and given any $z\in\mathbb{H}$, we have that $Tz=z+1$, so we will have that $z\sim z+1$ (and in general $z\sim z+n$ for any $n\in\mathbb{Z}$, so we identify these two points under the group action. And when you mention a fundamental set, if you are talking about a fundamental domain that is precisely what is meant by $\Gamma(1)\backslash \mathbb{H}$, so it is a set such that every element of $\mathbb{H}$ is equivalent to some element of $\Gamma(1)\backslash\mathbb{H}$, and if two elements in $\Gamma(1)\backslash\mathbb{H}$ are equivalent then they must lie on the boundary. It is a well-known theorem that you can look up (say in Serre's A Course in Arithmetic or other books) that you can this has a fundamental domain of $$\Gamma(1)\backslash\mathbb{H}=\{x+iy\in\mathbb{H}:\vert x\vert\leq \frac{1}{2},\vert x+iy\vert\geq 1\}$$ Thus, that is what is meant by $\Gamma(1)\backslash\mathbb{H}$, and then you can actually view this as a hyperbolic manifold (since $\mathbb{H}$ can by viewed as a hyperbolic space, and this is the quotient of hyperbolic space by a Lie group). This is sometimes referred to as the modular surface if you want to look more into it.

Answer 2

This refers to the set of orbits of points in $\mathcal{H}$ under the action of the modular group $\Gamma(1)$.

The image to have in mind is the fundamental domain, which is the shaded part in the following image (taken from wikipedia)

More specifically, the set of orbits can be represented by the points in the interior of the shaded part and a choice of "one half" of the points on the boundary.

The set of orbits can be thought of as a Riemann surface, and when compactified by adding a cusp at $\infty$ it becomes a compact Riemann surface. More generally, the set of orbits of the upper half-plane under any congruence sugroup, $\Gamma(N) \backslash \mathcal{H}$, can be similarly compactified by adding a finite number of cusps.

Answer 3

The relevant piece of AT is the covering theory, specifically, regular coverings; there you encounter the notation $X/\Gamma$ where $\Gamma$ is a group acting continuously on a topological space $X$. The space $X/\Gamma$ is the quotient space of $X$ by the $\Gamma$-action, equipped with the quotient topology.

The remaining mystery to resolve is the backslash notation favored by people studying lattices in Lie groups. Here is the story:

Suppose that $\Gamma$ is a discrete subgroup of a Lie group $G$ (e.g. $SL(2, {\mathbb R})$ as in your case). Consider a maximal compact subgroup $K< G$. The right quotient $G/K$ is naturally diffeomorphic to a certain homogeneous space (a Riemannian manifold) $X$. In your case, $K=SO(2)$, $X$ is the hyperbolic plane. Now, you are left with the left action of $\Gamma$ on $G$, hence, the left action of $\Gamma$ on $G/K=X$. How would one denote the quotient of $X$ by this left action of $\Gamma$? A topologist would simply write $X/\Gamma$, which translates into ugly and unclear $G/K/\Gamma$. Thus, people in your field, write instead $\Gamma\backslash X$ or, frequently, $\Gamma \backslash G/K$, where the backslash indicates the quotient by the left action of $\Gamma$.