Jordan curve, and if it is equal to the Jordan curve the group is said to be of the first kind, and otherwise it is said to be of the second kind. Schottky groups Let Ci be the boundary circles of a finite collection of disjoint closed disks. The group generated by inversion in each circle has limit set a Cantor set, and the quotient H3/G is a mirror orbifold with underlying space a ball. It is double covered by a handlebody; the corresponding index 2 subgroup is a Kleinian group called a Schottky group. Crystallographic groups Let T be a periodic tessellation of hyperbolic 3-space. The group of symmetries of the tessellation is a Kleinian group. Fundamental groups of hyperbolic 3-manifolds The fundamental group of any oriented hyperbolic 3-manifold is a Kleinian group. There are many examples of these, such as the complement of a figure 8 knot or the Seifert–Weber space. Conversely if a Kleinian group has no nontrivial torsion elements then it is the fundamental group of a hyperbolic 3-manifold. Degenerate Kleinian groups A Kleinian group is called degenerate if it is not elementary and its limit set is simply connected. Such groups can be constructed by taking a suitable limit of quasi-Fuchsian groups such that one of the two components of the regular points contracts down to the empty set; these groups are called singly degenerate. If both components of the regular set contract down to the empty set, then the limit set becomes a space-filling curve and the group is called doubly degenerate. The existence of degenerate Kleinian groups was first shown indirectly by , and the first explicit example was found by Jørgensen. gave examples of doubly degenerate groups and space-filling curves associated to pseudo-Anosov maps. See also Ahlfors measure conjecture Density theorem for Kleinian groups Ending lamination theorem Tameness theorem (Marden's conjecture) References External links A picture of the limit set of a quasi-Fuchsian group from . A picture of the limit set of a Kleinian group from . This was one of the first pictures of a limit set. A computer drawing of the same limit set Animations of Kleinian group limit sets Images related to Kleinian groups by McMullen Discrete groups Lie groups Automorphic forms 3-manifolds