Uniform perfectness of the limit sets of Kleinian groups

Sugawa, Toshiyuki

doi:10.1090/S0002-9947-01-02775-1

Uniform perfectness of the limit sets of Kleinian groups
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Trans. Amer. Math. Soc. 353 (2001), 3603-3615 Request permission

Abstract:

In this note, we show, in a quantitative fashion, that the limit set of a non-elementary Kleinian group is uniformly perfect if the quotient orbifold is of Lehner type, i.e., if the space of integrable holomorphic quadratic differentials on it is continuously contained in the space of (hyperbolically) bounded ones. This result covers the known case when the group is analytically finite. As applications, we present estimates of the Hausdorff dimension of the limit set and the translation lengths in the region of discontinuity for such a Kleinian group. Several examples will also be given.

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