Schottky group

Schottky group

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

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 2"g" disjoint Jordan curves "A"1, "B"1,..., "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.

References

*David Mumford, Caroline Series, and David Wright, "", Cambridge University Press, 2002 ISBN 0-521-35253-3
*Bernard Maskit, "Kleinian groups", Springer-Verlag, 1987 ISBN 0-387-17746-9


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