# Kleinian model

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Main article: [Beltrami–Klein model](/source/Beltrami%E2%80%93Klein_model)

In [mathematics](/source/Mathematics), a **Kleinian model** is a model of a three-dimensional [hyperbolic manifold](/source/Hyperbolic_manifold) *N* by the [quotient space](/source/Quotient_space_(topology)) \mathbb{H}^3 / \Gamma where \Gamma is a [discrete subgroup](/source/Discrete_group) of PSL(2,**C**). Here, the subgroup \Gamma, a [Kleinian group](/source/Kleinian_group), is defined so that it is isomorphic to the [fundamental group](/source/Fundamental_group) \pi_1(N) of the surface *N*.[1] Many authors use the terms *Kleinian group* and *Kleinian model* interchangeably, letting one stand for the other. The concept is named after [Felix Klein](/source/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](/source/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](/source/Isolated_point), of orientation-preserving [isometries](/source/Isometry) of [hyperbolic 3-space](/source/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.[2]

Many properties of Kleinian models are in direct analogy to those of [Fuchsian models](/source/Fuchsian_model);[3] however, overall, the theory is less well developed. A number of unsolved conjectures on Kleinian models are the analogs to theorems on Fuchsian models.[citation needed]

## See also

- [Hyperbolic 3-manifold](/source/Hyperbolic_3-manifold)

## References

1. Matsuzaki & Taniguchi 1998, pp. 27–30.

1. Elstrodt, Grunewald & Mennicke 1997, pp. 22–27.

1. Matsuzaki & Taniguchi 1998, pp. 68–71.

### Sources

- Matsuzaki, Katsuhiro & Taniguchi, Masahiko (1998). *Hyperbolic Manifolds and Kleinian Groups*. Clarendon Press. ISBN 0-19-850062-9.
- Elstrodt, Jürgen; Grunewald, Fritz; Mennicke, Jens (1997). *Groups Acting on Hyperbolic Space*. Springer. ISBN 3-540-62745-6.

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