M Theory Lesson 196

One place where the origin in the plane naturally appears in the theory of the modular group is in Reduction Theory, nicely explained in a paper recommended by Thomas Riepe. One relaxes the condition that the group action on the upper half plane shift the fundamental domain so that there is no intersection between the two domains, and allows a finite intersection. Then the region shown on the right, which is three times bigger than the usual domain, is allowed as a fundamental domain. Now let $\Gamma (N)$ be the congruence subgroup of the modular group. The translations of the new fundamental region give quotient spaces with punctures at vertices. For $\Gamma (3)$ one has a tetrahedron (like the neutrino tetrahedron), for $\Gamma (4)$ an octahedron and for $\Gamma (5)$ an icosahedron.