I hope you can help me by quickly explaining the following notation: $\Gamma \tau$. This notation is encountered in A First Course in Modular Forms by Fred Diamond and Jerry Shurman (love the book by the way). A modular curve is introduced as $Y(\Gamma) = \Gamma \backslash \mathcal{H} = \{ \Gamma \tau : \tau \in \mathcal{H} \}$ where $\mathcal{H}$ is the complex upper half plane. $\Gamma$ is a congruence subgroup of the modular group $SL_2(\mathbb{Z})$. So what does $\Gamma \tau$ mean? Does it mean all the elements $\gamma \tau$ with $\gamma \in \Gamma$, i.e., $\Gamma \tau = \{\gamma \tau : \gamma \in \Gamma\}$ for some fixed $\tau$?
I hope you can help me clarify this. It seems to be a very central concept. ;)
Thanks!