Quotient space by discrete group and $L^2(\Gamma\backslash \mathcal{H})$ I'm reading Daniel Bump's "Automorphic forms and representations"
chapter 2, and in the book they define an integral over $\Gamma\backslash\mathcal{H}$ (here, $\Gamma$ is a discontinuous subgroup of $\mathrm{SL}(2,\mathbb{R})$, and $\mathcal{H}$ is the upper half plane.)
But how to define it? what is the definition of that integral?
(or, is $\Gamma\backslash\mathcal{H}$ a manifold? how to prove it?)
If possible, please recommend a book or chapter of a book for elementary theory about $\Gamma\backslash\mathcal{H}$ when $\Gamma$ is a discrete subgroup of $\mathrm{SL}(2,\mathbb{R})$.
 A: In Section 1.2 of Bump’s book ‘The Modular Group’, you can see that Bump identifies $\mathcal{H}$ with the coset $SL_2(\mathbb{R})/ SO(2)$ by means of its Iwasawa decomposition.
In exercise 1.2.6 he gives away the Haar measure on $SL_2(\mathbb{R})$, and then pushing it forward to the coset, we get the measure on $\mathcal{H}$. This measure is invariant under $SL_2(\mathbb{R})$, so it is also invariant under any of its discontinuous subgroup ‘$\Gamma$’ , and so we can define the integral over $\Gamma \backslash \mathcal{H}$ as a quotient measure. (Mostly we are only concerned with $\Gamma$ such that $\Gamma \backslash \mathcal{H}$  has a finite volume)
Note that although I’m saying define, it’s more so using a few simple theorems from harmonic analysis to say that a such a measure exists and is unique up to scalar multiplication. Things are much simpler if $\Gamma$ is a subgroup of $SL_2(\mathbb{Z})$, where one can explicitly make use of the translates of the fundamental domain to construct $\Gamma \backslash \mathcal{H}$.
As for $\Gamma \backslash \mathcal{H}$ being a manifold, the same section 1.2 has the answer, that it can be made into a compact Riemann surface by adding a finite number of points (here also we require the finite volume of $\Gamma \backslash \mathcal{H}$), one point for each ‘cusps’ of $\Gamma$.
