Möbius transformation in the context of "Kleinian group"

In mathematics, a Kleinian group is a discrete subgroup of the group of orientation-preserving isometries of hyperbolic 3-space H. The latter, identifiable with PSL(2, C), is the quotient group of the 2 by 2 complex matrices of determinant 1 by their center, which consists of the identity matrix and its product by −1. PSL(2, C) has a natural representation as orientation-preserving conformal transformations of the Riemann sphere, and as orientation-preserving conformal transformations of the open unit ball B in R. The group of Möbius transformations is also related as the non-orientation-preserving isometry group of H, PGL(2, C). So, a Kleinian group can be regarded as a discrete subgroup acting on one of these spaces.