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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 $\mathcal{H}$).

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