Kleinian group

In mathematics, Kleinian groups play a central role in three -dimensional topology, hyperbolic geometry and complex analysis.

Definition

A Kleinian group is a discrete subgroup of, the isometry group of the 3-dimensional hyperbolic space.

A Kleinian group is called

• Torsion-free, if all elements have infinite order
• Not elementary if it is not virtually cyclic.

Hyperbolic manifold

If a torsion-free Kleinian group, then is a hyperbolic manifold, sometimes referred to as Klein manifold.

Limit set

The limit set of a Kleinian group Γ is a subset of the Riemann number sphere, defined as the average of the edge at infinity with the completion of a train Γx, where x is a point of hyperbolic space is. The definition of the limit set is independent of the chosen point x.

The now proven Ahlfors conjecture states that the limit set of a finitely generated Kleinian group is either completely or Lebesgue measure zero. ( The conjecture was proved by Canary 1993 for topologically tame groups. Together with the 2004 proven by Agol, Calegari and Gabai Tame PRIME guess it follows the validity for all finitely generated Kleinian groups. )

A Kleinian group is called Kleinian group first kind if the limit set is all about. Otherwise, it is a Kleinian group 2 Article

If a Kleinian group is second nature, then the hyperbolic manifold has infinite volume, in particular, it is then not compact.

Surface groups

It is a discrete, faithful representation of a surface group. Then the Klein group is called a Fuchsian group if its limit set is a circle Quasifuchssche group if its limit set is a Jordan curve and degenerate Kleinian group else a degenerate Kleinian group is called doubly degenerate if its limit set is the whole 2-sphere and easy degenerate if the complement of the limit set is connected and not empty.

Geometrically finite Kleinian groups

A Kleinian group is called geometrically finite if it satisfies one of the following equivalent conditions:

• There is a Fundamentalpolyeder with finitely many faces
• For all the Dirichlet area has finitely many faces
• The convex core of has finite volume.

One end of a hyperbolic 3-manifold is called geometrically finite if it has a neighborhood which is disjoint from the convex core. Otherwise, the end is called geometrically infinite.

An area group is geometrically finite if and only if it is a quasifuchssche group.

Geometric infinite ends

If one end of a hyperbolic 3-manifold is geometrically infinite, then there are with any environment of a closed geodesic. For a geometrically infinite end of the mold to define the Endelaminierung as the lamination of the surface, which gives a ( any ) sequence of each compact subset ultimately leaving geodesics as a limit.

The proven by Jeffrey Brock, Richard Canary and Yair Minsky ending lamination theorem states that geometrically infinite ends are uniquely determined by their Endelaminierung.