发表机构
University of Illinois at Urbana Champaign（伊利诺伊大学厄巴纳-香槟分校）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文通过对数重度量研究双曲群视觉边界，证明其与多项式Floyd边界双Lipschitz等价，保持Nagata维数和渐近维数，并检测单端性，但对数度量与视觉度量非拟对称等价。
AI 中文摘要
我们研究视觉边界的一种对数重度量，该度量源于Garg-Jana-Qing引入的次线性Morse边界上的对数度量。对于每一个真测地双曲空间$X$，我们证明$d_{\mathrm{log}}(\xi,\eta)\asymp (1+(\xi\mid\eta)_o)^{-\theta}$。作为推论，在边界的自然等同下，$d_{\mathrm{log}}$与径向密度$g_\theta(t)=(1+t)^{-1-\theta}$对应的多项式Floyd边界度量是双Lipschitz等价的。真测地双曲空间之间的每个拟等距都诱导其对数边界之间具有固定容许指数的双Lipschitz映射。对于每个非初等双曲群，对数边界是非倍数的，并且具有无限的Hausdorff维数和Assouad维数。然而，对数重度量保持Nagata（容量）维数，并继续恢复群的渐近维数。它还通过线性连通性（有界转向）检测单端性，并保持均匀完美性。因此，对数量度和视觉度量不是拟对称等价的，尽管若干粗几何特征在对数边界中仍然可见。
英文摘要
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.
Comments46 pages, 1 Figure
