I am trying to solve exercise 2.2.1 in Diamond and Shurman:
Let $\Gamma$ be a congruence subgroup of $SL_2(\mathbb{Z})$, and $\pi:\mathcal{H}\to Y(\Gamma)$ the quotient map from the complex upper half plane to the corresponding modular curve. Let $\tau\in\mathcal{H}$ be a point fixed only by the identity transformation from $\Gamma$, i.e. $\pm I$. Show that $\tau$ has a neighbourhood $U$ such that $\pi|_U : U\to \pi(U)$ is a homeomorphism.
The problem says to use Proposition 2.1.1, which is the following:
For $\tau_1,\tau_2\in\mathcal{H}$, there exists respective neighbourhoods $U_1$, $U_2$ such that for all $\gamma\in SL_2(\mathbb{Z})$, if $\gamma(U_1)\cap U_2\neq \emptyset$ then $\gamma(\tau_1)=\tau_2$.
I know that $\pi$ is surjective, continuous and open, so I just need to find a neighbourhood of $\tau$ on which it is injective. But I simply cannot find such a neighbourhood - all the applications of Proposition 2.1.1. I can think of requires me to intersect infinitely many neighbourhoods, so we aren't guaranteed the intersection is open. I do not think it is too hard, but for the life of me I cannot find the argument. Any hints appreciated.