Kleinian model
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In mathematics, a Kleinian model is a model of a three-dimensional hyperbolic manifold N by the quotient space <math>\mathbb{H}^3 / \Gamma</math> 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 <math>\pi_1(N)</math> of the surface N. Many authors use the terms Kleinian group and Kleinian model interchangeably, letting one stand for the other. See also hyperbolic 3-manifold.
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.
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