Definition of cusp of a congruence group

Question

I am reading p.22 of Dan Bump's Automorphihic forms and representations. A cusp of the congruence group acting on the upper half plane is defined to be an orbit of the action of the congruence subgroup on $\mathbb{Q} \mathbb{P}^1$. It also says that intuitively, the cusps are the places where a fundamental domain of the congruence subgroup touching the boundary of the upper half plane.

Question: Why does the definition mean what the intuition says?

Thanks!

Answer 1

You should identify the upper half plane with a subspace of $\mathbb{C}$ with a subspace of the Riemann sphere. In this identification $\mathbb{P}^1(\mathbb{R})$ is a great circle separating the upper half plane and lower half plane, and $\mathbb{P}^1(\mathbb{Q})$ is the orbit of $\infty$ under $\text{PSL}_2(\mathbb{Z})$. This orbit breaks up into a union of orbits under any congruence subgroup and these are the points at which a fundamental domain can touch the boundary because

• under the action of $\text{PSL}_2(\mathbb{Z})$ the fundamental domain touches the boundary only at $\infty$, and
• a fundamental domain for any congruence subgroup is a union of translates of fundamental domains for $\text{PSL}_2(\mathbb{Z})$.