In mathematics, a Schottky group is a special sort of Kleinian group, named after Friedrich Schottky. In mathematics, a Kleinian group is a finitely generated discrete group Γ of conformal (i. ... Friedrich Hermann Schottky (July 24, 1851 - August 12, 1935) was a German mathematician who worked on elliptic, abelian, and theta functions and invented Schottky groups. ...

Definition

Fix some point p on the Riemann sphere. Each Jordan curve not passing through p divides the Riemann sphere into two pieces, and we call the piece containing p the "exterior" of the curve, and the other piece its "interior". Suppose there are 2g disjoint Jordan curves A₁, B₁,..., A_g, B_g in the Riemann sphere with disjoint interiors. If there are Moebius transformations T_i taking the outside of A_i onto the inside of B_i, then the group generated by these transformations is a Kleinian group. A Schottky group is any Kleinian group that can be constructed like this. In topology, the Jordan curve theorem states that every non-self-intersecting loop in the plane divides the plane into an inside and an outside. It was proved by Oswald Veblen in 1905. ... A rendering of the Riemann Sphere. ... Möbius transformations should not be confused with the Möbius transform. ...

Schotty groups are finitely generated free groups such that all non-trivial elements are loxodromic. Conversely Maskit showed that any finitely generated free Kleininan group such that all non-trivial elements are loxodromic is a Schottky group. In abstract algebra, a generating set of a group is a subset S such that every element of G can be expressed as the product of finitely many elements of S and their inverses. ... The Cayley graph of the free group on two generators a and b In mathematics, a group G is called free if there is a subset S of G such that any element of G can be written in one and only one way as a product of finitely many... In mathematics, a Möbius transformation is a bijective conformal mapping of the extended complex plane (i. ...

References

• David Mumford, Caroline Series, and David Wright, Indra's Pearls: The Vision of Felix Klein, Cambridge University Press, 2002 ISBN 0-521-35253-3
• Bernard Maskit, Kleinian groups, Springer-Verlag, 1987 ISBN 0-387-17746-9