TY - JOUR

T1 - Uniform perfectness of the limit sets of Kleinian groups

AU - Sugawa, Toshiyuki

PY - 2001

Y1 - 2001

N2 - 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.

AB - 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.

KW - Hausdorff dimension

KW - Kleinian group

KW - Translation length

KW - Uniformly perfect

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U2 - 10.1090/s0002-9947-01-02775-1

DO - 10.1090/s0002-9947-01-02775-1

M3 - Article

AN - SCOPUS:23044525259

SN - 0002-9947

VL - 353

SP - 3603

EP - 3615

JO - Transactions of the American Mathematical Society

JF - Transactions of the American Mathematical Society

IS - 9

ER -