Kleinian Group - Definitions

Definitions

By considering the ball's boundary, a Kleinian group can also be defined as a subgroup Γ of PGL(2,C), the complex projective linear group, which acts by Möbius transformations on the Riemann sphere. 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.

When Γ is isomorphic to the fundamental group of a hyperbolic 3-manifold, then the quotient space H3/Γ becomes a Kleinian model of the manifold. Many authors use the terms Kleinian model and Kleinian group interchangeably, letting the one stand for the other.

Discreteness implies points in B3 have finite stabilizers, and discrete orbits under the group G. But the orbit Gp of a point p will typically accumulate on the boundary of the closed ball .

The boundary of the closed ball is called the sphere at infinity, and is denoted . The set of accumulation points of Gp in is called the limit set of G, and usually denoted . The complement is called the domain of discontinuity or the ordinary set or the regular set. Ahlfors' finiteness theorem implies that if the group is finitely generated then is a Riemann surface orbifold of finite type.

The unit ball B3 with its conformal structure is the Poincaré model of hyperbolic 3-space. When we think of it metrically, with metric

it is a model of 3-dimensional hyperbolic space H3. The set of conformal self-maps of B3 becomes the set of isometries (i.e. distance-preserving maps) of H3 under this identification. Such maps restrict to conformal self-maps of, which are Möbius transformations. 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).