I noticed this notation in a mathematical paper and it wasn't defined, so I assume it must be standard. Let $\Gamma_0(N)$ be the congruence subgroup $$\left\{\pmatrix{a & b \\ c & d} : ad - bc = 1, c \equiv 0 \pmod N\right\}$$ of the modular group $\Gamma = \textrm{SL}_2(\mathbb{Z})$. The notation I encountered was $$\Gamma_{\infty} \backslash \Gamma_0(N).$$ My understanding is that this type of notation usually refers to the set of equivalence classes obtained from action of the group to the left of the slash on the set to the right of the slash, i.e. $\textrm{SL}_2(\mathbb{Z}) \backslash \mathbb{H}$. However, I'm not sure what $\Gamma_{\infty}$ is supposed to mean.
Congruence subgroup action notation
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$\begingroup$
notation
modular-forms
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0$\Gamma_{\infty}$ is the stabiliser of the cusp of infinity, namely the subgroup of $\Gamma_0(N)$ consisting of matrices of the form $\pm \begin{pmatrix} 1 & n \\ 0 & 1 \end{pmatrix}$ with $n \in \mathbb{Z}$. – 2017-01-29
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0Ah, okay, great, thanks. So is that notation referring to the orbits of the action (left multiplication) of $\Gamma_{\infty}$ on the set $\Gamma_0(N)$, or is it supposed to be the quotient group $\Gamma_0(N) / \Gamma_{\infty}$? – 2017-03-01
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0The former. Quotient groups only make sense when the subgroup is normal, which is not the case here. – 2017-03-01