Schottky group

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In mathematics, a Schottky group is a special sort of Kleinian group, named after Friedrich Schottky.

[edit] 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.

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.

[edit] 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