A new conformal gauge on the visual boundary of a hyperbolic group

Abstract

We study a logarithmic re-gauging of the visual boundary, arising from the logarithmic metric on the sublinear Morse boundary introduced by Garg-Jana-Qing. For every proper geodesic hyperbolic space $X$, we prove that $d_{\mathrm{log}}(ξ,η)\asymp (1+(ξ\midη)_o)^{-θ}$. As a consequence, under the natural identification of boundaries, $d_{\mathrm{log}}$ is bi-Lipschitz equivalent to the polynomial Floyd boundary metric associated with the radial density $g_θ(t)=(1+t)^{-1-θ}$. Every quasi-isometry between proper geodesic hyperbolic spaces induces a bi-Lipschitz map between their logarithmic boundaries with a fixed admissible exponent. For every non-elementary hyperbolic group, the logarithmic boundary is nondoubling and has infinite Hausdorff and Assouad dimensions. Nevertheless, the logarithmic re-gauging preserves Nagata (capacity) dimension and continues to recover the asymptotic dimension of the group. It also detects one-endedness through linear connectedness (bounded turning) and preserves uniform perfectness. Thus the logarithmic and visual metrics are not quasisymmetrically equivalent, although several coarse-geometric features remain visible in the logarithmic boundary.