of Γ, 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). Variations 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. Types A Kleinian group is said to be of finite type if its region of discontinuity has a finite number of orbits of components under the group action, and the quotient of each component by its stabilizer is a compact Riemann surface with finitely many points removed, and the covering is ramified at finitely many points. A Kleinian group is called finitely generated if it has a finite number of generators. The Ahlfors finiteness theorem says that such a group is of finite type. A Kleinian group Γ has finite covolume if H3/Γ has finite volume. Any Kleinian group of finite covolume is finitely generated. A Kleinian group is called geometrically finite if it has a fundamental polyhedron (in hyperbolic 3-space) with finitely many sides. Ahlfors showed that if the limit set is not the whole Riemann sphere then it has measure 0. A Kleinian group Γ is called arithmetic if it is commensurable with the group norm 1 elements of an order of quaternion algebra A ramified at all real places over a number field k with exactly one complex place. Arithmetic Kleinian groups have finite covolume. A Kleinian group Γ