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I read about how Hecke groups are a particular generalisation of the modular group, generalising one of the generators of the modular group to $z\mapsto z + \lambda$. Have people studied generalisations of the other generator of the modular group, perhaps generalising to $z \mapsto \dfrac{-1}{z^n}$ ?

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    It would be $z \mapsto \sqrt[n]{\dfrac{-1}{z}}$, but the roots in the upper-half plane would all have to be identified.2011-02-25

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