In mathematics, a Kleinian model is a model of a three-dimensional hyperbolic manifold N by the quotient space where Γ is a discrete subgroup of PSL(2,C). Here, the subgroup Γ, a Kleinian group, is defined so that it is isomorphic to the fundamental group π₁(N) of the surface N. Many authors use the terms Kleinian group and Kleinian model interchangeably, letting one stand for the other. Mathematics, often abbreviated maths in Commonwealth English and math in American English, is the study of abstraction. ... For quotient spaces in linear algebra, see quotient space (linear algebra). ... In mathematics, a Kleinian group is a finitely generated discrete group Γ of conformal (i. ... In mathematics, the fundamental group is one of the basic concepts of algebraic topology. ...

Many properties of Kleinian models are in direct analogy to those of Fuchsian models; however, overall, the theory is less well developed. A number of unsolved conjectures on Kleinian models are the analogs to theorems on Fuchsian models. In mathematics, a Fuchsian model is a representation of a hyperbolic Riemann surface R. By the uniformization theorem, every Riemann surface is either elliptic, parabolic or hyperbolic. ...