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 . A Kleinian group is said to be of type 1 if the limit set is the whole Riemann sphere, and of type 2 otherwise. Examples The Maskit slice through the moduli space of Kleinian groups Bianchi groups A Bianchi group is a Kleinian group of the form PSL(2, Od), where is the ring of integers of the imaginary quadratic field for d a positive square-free integer. Elementary and reducible Kleinian groups A Kleinian group is called elementary if its limit set is finite, in which case the limit set has 0, 1, or 2 points. Examples of elementary Kleinian groups include finite Kleinian groups (with empty limit set) and infinite cyclic Kleinian groups. A Kleinian group is called reducible if all elements have a common fixed point on the Riemann sphere. Reducible Kleinian groups are elementary, but some elementary finite Kleinian groups are not reducible. Fuchsian groups Any Fuchsian group (a discrete subgroup of PSL(2, R)) is a Kleinian group, and conversely any Kleinian group preserving the real line (in its action on the Riemann sphere) is a Fuchsian group. More generally, every Kleinian group preserving a circle or straight line in the Riemann sphere is conjugate to a Fuchsian group. Koebe groups A factor of a Kleinian group G is a subgroup H maximal subject to the following properties: H has a simply connected invariant component D A conjugate of an element h of H by a conformal bijection is parabolic or elliptic if and only if h is. Any parabolic element of G fixing a boundary point of D is in H. A Kleinian group is called a Koebe group if all its factors are elementary or Fuchsian. Quasi-Fuchsian groups A Kleinian group that preserves a Jordan curve is called a quasi-Fuchsian group. When the Jordan curve is a circle or a straight line these are just conjugate to Fuchsian groups under conformal transformations. Finitely generated quasi-Fuchsian groups are conjugate to Fuchsian groups under quasi-conformal transformations. The limit set is contained in the invariant