Schottky group
In mathematics, a Schottky group is a special sort of Kleinian group, first studied by Friedrich Schottky (1877).
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,..., Ag, Bg in the Riemann sphere with disjoint interiors. If there are Möbius transformations Ti taking the outside of Ai onto the inside of Bi, then the group generated by these transformations is a Kleinian group. A Schottky group is any Kleinian group that can be constructed like this.
By work of Maskit (1967), a Kleinian group is Schottky if and only if it is finitely generated, free, has nonempty domain of discontinuity, and all non-trivial elements are loxodromic.
A fundamental domain for the action of a Schottky group G on its regular points Ω(G) in the Riemann sphere is given by the exterior of the Jordan curves defining it. The corresponding quotient space Ω(G)/G is given by joining up the Jordan curves in pairs, so is a compact Riemann surface of genus g. This is the boundary of the 3-manifold given by taking the quotient (H∪Ω(G))/G of 3-dimensional hyperbolic H space plus the regular set Ω(G) by the Schottky group G, which is a handlebody of genus g. Conversely any compact Riemann surface of genus g can be obtained from some Schottky group of genus g.
A Schottky group is called classical if all the disjoint Jordan curves corresponding to some set of generators can be chosen to be circles. Marden (1974, 1977) gave an indirect and non-constructive proof of the existence of non-classical Schottky groups, and Yamamoto (1991) gave an explicit example of one. It has been shown by Doyle (1988) that all finitely generated classical Schottky groups have limit sets of Hausdorff dimension bounded above strictly by a universal constant less than 2. Conversely, Hou (2010) has proved that there exists a universal lower bound on the Hausdorff dimension of limit sets of all non-classical Schottky groups.
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