Kleinian model

Kleinian model

In mathematics, a Kleinian model is a model of a three-dimensional hyperbolic manifold "N" by the quotient space mathbb{H}^3 / Gamma 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 pi_1(N) 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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