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 dlog(ξ,η)asymp (1+(ξ|η)o)−θ. As a consequence, under the natural identification of boundaries, dlog is bi-Lipschitz equivalent to the polynomial Floyd boundary metric associated with the radial density gθ(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.
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BibTeX
@misc{garg2026conformal,
title = {{A new conformal gauge on the visual boundary of a hyperbolic group}},
author = {Manisha Garg},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.25468},
eprint = {2609.25468},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.25468},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - A new conformal gauge on the visual boundary of a hyperbolic group
AU - Manisha Garg
PY - 2026
DA - 2026/09/21
PB - arXiv
DO - 10.48550/arxiv.2609.25468
UR - https://arxiv.org/abs/2609.25468
AB - 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_log(ξ,η)asymp (1+(ξ|η)_o)^−θ. As a consequence, under the natural identification of boundaries, d_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.
ER -