Kleinian Group - Finiteness Conditions

Finiteness Conditions

• A Kleinian group is said to be of finite type if its region of discontinuity has a finite number of components, 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 is 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 of units 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 is called cocompact if H3/Γ is compact, or equivalently SL(2, C)/Γ is compact. Cocompact Kleinian groups have finite covolume.

• A Kleinian group is called topologically tame if it is finitely generated and its hyperbolic manifold is homeomorphic to the interior of a compact manifold with boundary.
• A Kleinian group is called geometrically tame if its ends are either geometrically finite or simply degenerate (Thurston 1980).