Let $\mathscr F$ be a fundamental domain for $SL_2(\Bbb Z)$ and let $\Gamma$ be a congruence subgroup with $-I \in \Gamma$. If $g_1,...,g_r$ are coset representatives of $\Gamma$ such that $\mathscr D=\cup_{i=1}^rg_i^{-1}\mathscr F$ has a connected interior, then $\mathscr D$ is a fundamental domain for $\Gamma$.
Here I am facing problem to show that no two interior point of $\mathscr D$ is $\Gamma$ equivalent.
Here suppose $z_1,z_2 \in int(\mathscr D)$ $\exists \gamma $ s.t $z_2=\gamma z_1$ and $\exists g_1, g_2 \in SL_2(\Bbb Z)$ s.t $g_1z_1,g_2z_2 \in \mathscr F$. Now if one of the $g_1z_1,g_2z_2 $ belongs to $int(\mathscr F)$. Then we'll arrive at contradiction. But if this happens that both $g_1z_1,g_2z_2 \in Bd(\mathscr F)$. Then how shall I argue? Or in other language how can I show that it could not be the case.