The modular group and fundamental domains

A matrix γ with entries a,b,c,d and determinant one acts by (aτ+b)/(cτ+d). Its denominator cannot vanish in the upper half-plane: a zero would put τ on the real axis, unless c=0, when d is nonzero. Multiplying numerator and denominator by the conjugate denominator gives the new imaginary part y divided by |cτ+d| squared. The action therefore preserves the upper half-plane and its hyperbolic metric, although it usually changes Euclidean lengths and heights.

The matrices γ and −γ induce the same map. The kernel is precisely the two scalar matrices ±I, so the faithful action is by PSL₂(ℤ), the quotient of SL₂(ℤ) by that kernel. With S=[[0,−1],[1,0]] and T=[[1,1],[0,1]], the maps are inversion with a minus sign and unit translation. In SL₂(ℤ), S squared and (ST) cubed equal −I; their images have orders two and three in PSL₂(ℤ). Group orders must be stated in the group actually being used.

Take the closed region defined by |x|≤1/2 and |τ|≥1. Every orbit meets it, and no two distinct points in its interior are equivalent. Boundary points still require identification: translation pairs the two vertical edges, and S pairs the two halves of the unit-circle arc. Calling the closed region a set of unique representatives without these qualifications counts some lattice shapes twice. A half-open boundary convention can remove duplication, but must handle the special vertices consistently.

A practical reduction alternates translation into the vertical strip with S whenever the point lies inside the unit circle. Inversion then increases its height. Termination is not a consequence of an arbitrary increasing real sequence; discreteness of the possible denominators supplies the missing argument. The domain theorem and generation by S,T are structural results, while explicit examples can be checked directly. On the quotient, the identified region has a noncompact end extending upward, despite having finite hyperbolic area.

The point i is fixed by S. The point ρ=exp(2πi/3), on the left circular endpoint, is fixed by ST. Their effective stabilizers have orders two and three. In the upper-half-plane covering, a local quotient coordinate consequently behaves like the square or cube of a local coordinate. The coarse quotient is still a Riemann surface; the stabilizer information explains its ramified uniformizing map and later fractional contributions in the valence formula. It is not a singularity of the upper half-plane itself.

The rational boundary points together with infinity form the cusp set. For the full modular group every rational cusp is equivalent to infinity, because a reduced fraction can be completed to a determinant-one matrix. Infinity is an end rather than a point of the original domain. Adding that cusp compactifies the quotient, with exponential coordinate q=exp(2πiτ) near the added point. This local coordinate identifies unit translations, while its vanishing records escape to infinite height. It will impose a genuinely additional holomorphy condition on modular forms.