Schottky group

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.

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.

That area is bounded above by a constant times the contribution to the Poincaré sum of elements of word length n, so decreases to 0.