Kleinian model

{{main|Beltrami–Klein model}}

In mathematics, a Kleinian model is a model of a three-dimensional hyperbolic manifold N by the quotient space $\backslash mathbb\{H\}^3\; /\; \backslash Gamma$ where $\backslash Gamma$ is a discrete subgroup of PSL(2,C). Here, the subgroup $\backslash Gamma$, a Kleinian group, is defined so that it is isomorphic to the fundamental group $\backslash pi\_1(N)$ of the surface N.{{sfn|Matsuzaki|Taniguchi|1998|pp= 27-30}} Many authors use the terms Kleinian group and Kleinian model interchangeably, letting one stand for the other. The concept is named after Felix Klein.

In less technical terms, a Kleinian model it is a way of assigning coordinates to a hyperbolic manifold, or a three-dimensional space in which every point locally resembles hyperbolic space. A Kleinian model is created by taking three-dimensional hyperbolic space and treating two points as equivalent if and only if they can be reached from each other by applying a member of a group action of a Kleinian group on the space. A Kleinian group is any discrete subgroup, consisting only of isolated points, of orientation-preserving isometries of hyperbolic 3-space. The group action of a group is a set of functions on a set which, roughly speaking, have the same structure as a group.{{sfn|Elstrodt|Grunewald|Mennicke|1997|pp= 22-27}}

Many properties of Kleinian models are in direct analogy to those of Fuchsian models;{{sfn|Matsuzaki|Taniguchi|1998|pp= 68-71}} however, overall, the theory is less well developed. A number of unsolved conjectures on Kleinian models are the analogs to theorems on Fuchsian models.{{cn|date=March 2025}}