Toshiyuki SUGAWA

Uniform perfectness of the limit sets of Kleinian groups


A compact set C in the Riemann sphere is said to be uniformly perfect if bounded are the moduli of those annuli in the complement which separate C. The limit set of an analytically finite non-elementary Kleinian group is known to be uniformly perfect.
In this note, we shall show that the limit set of a non-elementary Kleinian group is uniformly perfect if the quotient orbifold is of Lehner type, i.e., satisfies that the space of integrable holomorphic quadratic differentials on it is continuously contained in the space of (hyperbolically) bounded ones. Indeed, we shall state the theorem in more precise and quantitative form.
As applications, we present estimates of the Hausdorff dimension of the limit set and the translation length in the region of discontinuity.

submission: August 4, 1997
revision: August 7, 1998

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