Kleinian group

Algebraic structure → Group theory Group theory
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In mathematics, a Kleinian group is a discrete subgroup of the group of orientation-preserving isometries of hyperbolic 3-space H³. 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 B³ in R³. The group of Möbius transformations is also related as the non-orientation-preserving isometry group of H³, PGL(2, C). So, a Kleinian group can be regarded as a discrete subgroup acting on one of these spaces.