The upper half plane H is the set of complex numbers x+iy such that y>0. It is the domain of many functions of interest in complex analysis, especially modular forms. It is also a model of the hyperbolic plane.

There is a group action of SL₂(R) on H defined by

The action is transitive and the stabilizer of i is the rotation group

Therefore, H = SL₂(R)/K.