In
mathematics
Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...
, a Kleinian model is a model of a three-dimensional
hyperbolic manifold ''N'' by the
quotient space where
is a
discrete subgroup of PSL(2,C). Here, the subgroup
, a
Kleinian group
In mathematics, a Kleinian group is a discrete subgroup of the group (mathematics), group of orientation-preserving Isometry, isometries of hyperbolic 3-space . The latter, identifiable with PSL(2,C), , is the quotient group of the 2 by 2 complex ...
, is defined so that it is isomorphic to the
fundamental group
In the mathematics, mathematical field of algebraic topology, the fundamental group of a topological space is the group (mathematics), group of the equivalence classes under homotopy of the Loop (topology), loops contained in the space. It record ...
of the surface ''N''. Many authors use the terms ''Kleinian group'' and ''Kleinian model'' interchangeably, letting one stand for the other. The concept is named after
Felix Klein
Felix Christian Klein (; ; 25 April 1849 – 22 June 1925) was a German mathematician and Mathematics education, mathematics educator, known for his work in group theory, complex analysis, non-Euclidean geometry, and the associations betwe ...
.
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
In mathematics, hyperbolic space of dimension ''n'' is the unique simply connected, ''n''-dimensional Riemannian manifold of constant sectional curvature equal to −1.
It is homogeneous, and satisfies the stronger property of being a symme ...
. 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
In mathematics, an isometry (or congruence, or congruent transformation) is a distance-preserving transformation between metric spaces, usually assumed to be bijective. The word isometry is derived from the Ancient Greek: ἴσος ''isos'' mea ...
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.
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.
See also
*
Hyperbolic 3-manifold
In mathematics, more precisely in topology and differential geometry, a hyperbolic 3-manifold is a manifold of dimension 3 equipped with a hyperbolic metric, that is a Riemannian metric which has all its sectional curvatures equal to −1. It ...
References
Sources
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Hyperbolic geometry
Kleinian groups
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