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

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 nat