Uniform perfectness of the limit sets of Kleinian groups (Q2716152)

In this paper, the author shows that the limit set of a non-elementary Kleinian group is uniformly perfect provided the quotient orbifold is of Lehner type, that is, provided the space of integrable holomorphic quadratic differentials on it is continuously contained in the space of (hyperbolically) bounded ones. This condition generalizes previously known conditions. Uniform perfectness is related to various other quantities involving the geometry of the quotient surface, and the author uses this result to obtain estimates on the Hausdorff dimension of the limit set and on the translation lengths in the region of discontinuity of the Kleinian group, in terms of the multiplier of a loxodromic element. The paper contains several explicit examples, including an infinitely generated Kleinian group whose limit set is not uniformly perfect.