Actions on CAT(-1) spaces with critical exponent less than 1

Beibei Liu; Shi Wang

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arXiv · 2608.18906

Actions on CAT(-1) spaces with critical exponent less than 1

Beibei Liu·Shi Wang

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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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Beibei Liu, Shi Wang. 2026-08-19. Actions on CAT(-1) spaces with critical exponent less than 1. https://arxiv.org/abs/2608.18906

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