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Kleinian GroupIn mathematics, a Kleinian group is a finitely generated discrete group Γ of conformal (i.e. angle-preserving) self-maps of the open unit ball in . By considering the ball's boundary, a Kleinian group can also be defined as a subgroup Γ of PGL2(C), the complex projective linear group, whose action by Mbius transformations at some point of the Riemann sphere is freely discontinuous. When Γ is isomorphic to the fundamental group of a three-dimensional hyperbolic manifold, then the quotient space 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 have finite stabilizers, and discrete orbits under the group . But the orbit of a point 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 , and usually denoted . The unit ball with its conformal structure is the Poincare model of hyperbolic 3-space. When we think of it metrically, it is denoted . The set of conformal self-maps of becomes the set of isometries (i.e. distance-preserving maps) of under this identification. Such maps restrict to conformal self-maps of , which are Mobius transformations. There are isomorphisms -
Mob(S^2_\infty) \cong Conf(B^3) \cong Isom(H^3) The subgroups of these groups consisting of orientation-preserving transformations are all isomorphic to the matrix group -
via the usual identification of the unit sphere with the complex projective line . Example Reflection groups. Let be the boundary circles of a finite collection of disjoint closed disks. The group generated by inversion in each circle is a Kleinian group. The limit set is a Cantor set, and the quotient is a mirror orbifold with underlying space a ball. It is double covered by a handlebody; the corresponding index 2 subgroup is a Schottky group. Example Crystallographic groups. Let be a periodic tessellation of hyperbolic 3-space. The group of symmetries of the tessellation is a Kleinian group. Metric The canonical hyperbolic metric on the unit ball is given by -
for . References - Bernard Maskit, Kleinian Groups, (1988) Springer-Verlag New York ISBN 0-387-17746-9
- Katsuhiko Matsuzaki and Masahiko Taniguchi, Hyberbolic Manifolds and Kleinian Groups, (1998) Clarendon Press, Oxford ISBN 0-19-850062-9
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