Cusps in the Compactification of Modular Curves $Y(\Gamma)$ I'm currently reading some of the geometric theory behind the theory of modular forms, Diamond and Shurman's book is my main reference. If $\Gamma$ is a congruence subgroup of $SL_2(\mathbb{Z})$, the modular curve $Y(\Gamma)$ is defined to be the quotient $\Gamma \backslash \mathfrak{H}$. In order to compactify this curve one needs to add a finite number of cusps $s_1,\dots, s_n$ which are the distinct orbits of $\Gamma$ acting on $\mathbb{Q}\cup \{ \infty\}$.
The case where $\Gamma=SL_2(\mathbb{Z})$ makes sense to me since the group acts transivity on $\mathbb{Q}\cup \infty$, but for general $\Gamma$, why do we need to add in the other orbits (apart from the orbit containing $\infty$) in order to compactify the curve? 
A somewhat related question: Diamond and Shurman describe a modular form of weight $k$ for $\Gamma$ as being holomorphic at the "limit points" and yet I cannot find any other reference to this idea. How does one realize these cusps as "limit points"?
Any help would be appreciated!
 A: For the first question, you might find this elementary example helpful:
For $ p $ a prime, the modular curve $ X_{0}(p) = \mathcal{H} / \Gamma_{0} $
has two cusps on $ \mathbb{P}(\mathbb{Q}) $ which can be denoted by
$ [1,0] $ and $ [0,1] $. To see this, we partition rational numbers $ \mathbb{Q} $ as $ \{ t \in \mathbb{Q} : p|deno(t) \} \cup 
(\{ t \in \mathbb{Q}: p \not | deno(t) \} \backslash \{ 0 \})  $.
Here by $ deno(t) $ I meant denominator of t.
For the first set, let $ r = u/v $ where $ p | v $. Because of the irreducibility of the representation of $ r $, there must be $ c, d \in \mathbb{Z} $ such that $ cu - dv = 1 $. The matrix $ M=(-c,d;v,-u) \in \Gamma_{0}$ takes $ r $ to $ \infty $, i.e.,
$ \frac {-cr + d}{vr - u} = \infty $. Similarly for a given $ s = u/v $ you can always find a matrix in $ \Gamma_{0} $ taking $ \infty $ to $ s $.
For the second set, let $ r = u/v $ where $ p \not |v $. From $ gcd(u,v)=1$
and $ gcd(p,v)=1 $ we have that $ gcd(v, pu)=1 $ therefore, we have another
determinant-form expression $ cpu - dv = 1 $ for some $ c,d \in \mathbb{Z} $.
The matrix $ M=(v,-u;cp,-d) \in \Gamma_{0} $ is the matrix taking $ r $ to zero
and similarly for a given $ s=u/v $ you can find a matrix in $ \Gamma_{0} $ taking a given $ 0 $ to $ s $. it simply shows that the modular curve $ X_{0}(p) $ for a prime number $ p $ has two cusps which can be roughly represented by $ 0 $ and $ \infty $.
