Actions on CAT(-1) spaces with critical exponent less than 1
Beibei Liu, Shi Wang
Abstract
We show that for a discrete isometry subgroup acting on a proper CAT(-1) space X, if the critical exponent is less than 1, then the critical exponent equals the Hausdorff dimension of the entire limit set. Consequently, the limit set must be a Cantor set. As an application, we prove that any finitely generated, torsion-free discrete subgroup in Isom(X) with critical exponent less than one must be geometrically finite and free. This answers a question of Kapovich.
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