Hausdorff dimension of boundaries of relatively hyperbolic groups

TL;DR

This paper investigates the Hausdorff dimension of boundaries in relatively hyperbolic groups, showing it correlates with the group's growth rate and identifying key conical points influencing this dimension.

Contribution

It establishes the Hausdorff dimension of Floyd and Bowditch boundaries as proportional to the group's growth rate, introducing uniformly conical points and Alhfors regular sets.

Findings

Hausdorff dimension equals a constant times the group's growth rate

Existence of Alhfors regular sets with dimension approaching the Hausdorff dimension

Presence of uniformly conical points in the boundaries

Abstract

In this paper, 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 respectively, they are 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.