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Compare say to hecke subgroups which are congruence subgroups where we say the equivalence classes are given by the points where the fundamental domain intercepts the real axis as well as infinity.

Definition of a cusp :

A pointed end where two curves meet.

So my stupid question is, the fundamental domain for sl2(z), isn't ##t= e^{\pm\frac{2\pi}{3}}## such a point ? Isn't this a cusp ?

Thanks