The **genus** of a finite index subgroup $G$ of the modular group is the genus of the corresponding compact Riemann surface arising from the compactification of $G\backslash\mathcal{H},$ which is obtained by adjoining the cusps of $G$.

Here $\mathcal{H}$ denotes the complex upper halfplane on which $G$ acts by linear fractional transformations, and $G\backslash \mathcal{H}$ denotes the quotient of $\mathcal{H}$ by this action (points of $G\backslash\mathcal{H}$ correspond to $G$-orbits in $\mathcal{H}$).

**Authors:**

**Knowl status:**

- Review status: reviewed
- Last edited by Andrew Sutherland on 2018-06-20 13:25:58

**Referred to by:**

Not referenced anywhere at the moment.

**History:**(expand/hide all)

- 2018-06-20 13:25:58 by Andrew Sutherland (Reviewed)