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Hausdorff dimension of boundaries of relatively hyperbolic groups

Leonid Potyagailo and Wen-yuan Yang

Geometry & Topology 23 (2019) 1779–1840

We study the Hausdorff dimension of the Floyd and Bowditch boundaries of a relatively hyperbolic group, and show that, for the Floyd metric and shortcut metrics, they are both equal to a constant times the growth rate of the group.

In the proof, we study a special class of conical points called uniformly conical points and establish that, in both boundaries, there exists a sequence of Alhfors regular sets with dimension tending to the Hausdorff dimension and these sets consist of uniformly conical points.

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Floyd boundary, Hausdorff dimension, growth rate, conical points, Ahlfors-regular
Mathematical Subject Classification 2010
Primary: 20F65
Secondary: 20F67
Received: 28 November 2016
Revised: 13 June 2018
Accepted: 7 October 2018
Published: 17 June 2019
Proposed: Jean-Pierre Otal
Seconded: Dmitri Burago, Leonid Polterovich
Leonid Potyagailo
UFR de Mathématiques
Université de Lille 1
Villeneuve d’Ascq
Wen-yuan Yang
Beijing International Center for Mathematical Research & School of Mathematical Sciences
Peking University