Modular Group - Hecke Groups

Hecke Groups

The modular group can be generalized to the Hecke groups, named for Erich Hecke, and defined as follows.

The Hecke group Hq is the discrete group generated by

and where

The modular group Γ is isomorphic to H3 and they share properties and applications – for example, just as one has the free product of cyclic groups

more generally one has

which corresponds to the triangle group (2,q,∞). There is similarly a notion of principal congruence subgroups associated to principal ideals in For small values of q, one has:

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