Kleinian group

The latter, identifiable with PSL(2, C), is the quotient group of the 2 by 2 complex matrices of determinant 1 by their center, which consists of the identity matrix and its product by −1.

PSL(2, C) has a natural representation as orientation-preserving conformal transformations of the Riemann sphere, and as orientation-preserving conformal transformations of the open unit ball B3 in R3.

So, a Kleinian group can be regarded as a discrete subgroup acting on one of these spaces.

Hyperbolic 3-space has a natural boundary; in the ball model, this can be identified with the 2-sphere.

A hyperbolic isometry extends to a conformal homeomorphism of the sphere at infinity (and conversely, every conformal homeomorphism on the sphere at infinity extends uniquely to a hyperbolic isometry on the ball by Poincaré extension.

It is a standard result from complex analysis that conformal homeomorphisms on the Riemann sphere are exactly the Möbius transformations, which can further be identified as elements of the projective linear group PGL(2,C).

Thus, a Kleinian group can also be defined as a subgroup Γ of PGL(2,C).

Classically, a Kleinian group was required to act properly discontinuously on a non-empty open subset of the Riemann sphere, but modern usage allows any discrete subgroup.

of a hyperbolic 3-manifold, then the quotient space H3/Γ becomes a Kleinian model of the manifold.

On the other hand, the orbit Γp of a point p will typically accumulate on the boundary of the closed ball

is called the limit set of Γ, and usually denoted

The unit ball B3 with its conformal structure is the Poincaré model of hyperbolic 3-space.

There are isomorphisms The subgroups of these groups consisting of orientation-preserving transformations are all isomorphic to the projective matrix group: PSL(2,C) via the usual identification of the unit sphere with the complex projective line P1(C).

There are some variations of the definition of a Kleinian group: sometimes Kleinian groups are allowed to be subgroups of PSL(2, C).2 (that is, of PSL(2, C) extended by complex conjugations), in other words to have orientation reversing elements, and sometimes they are assumed to be finitely generated, and sometimes they are required to act properly discontinuously on a non-empty open subset of the Riemann sphere.

A Kleinian group is called reducible if all elements have a common fixed point on the Riemann sphere.

When the Jordan curve is a circle or a straight line these are just conjugate to Fuchsian groups under conformal transformations.

Let Ci be the boundary circles of a finite collection of disjoint closed disks.

A Kleinian group is called degenerate if it is not elementary and its limit set is simply connected.

The existence of degenerate Kleinian groups was first shown indirectly by Bers (1970), and the first explicit example was found by Jørgensen.

Cannon & Thurston (2007) gave examples of doubly degenerate groups and space-filling curves associated to pseudo-Anosov maps.