Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · 2007.11143

Uniformizing Gromov hyperbolic spaces with Busemann functions

Clark Butler
View original source ↗Read source document ↗Export metadata ↓

Abstract

Given a complete Gromov hyperbolic space $X$ that is roughly starlike from a point $ω$ in its Gromov boundary $\partial_{G}X$, we use a Busemann function based at $ω$ to construct an incomplete unbounded uniform metric space $X_{\varepsilon}$ whose boundary $\partial X_{\varepsilon}$ can be canonically identified with the Gromov boundary $\partial_ωX$ of $X$ relative to $ω$. This uniformization construction generalizes the procedure used to obtain the Euclidean upper half plane from the hyperbolic plane. Furthermore we show, for an arbitrary metric space $Z$, that there is a hyperbolic filling $X$ of $Z$ that can be uniformized in such a way that the boundary $\partial X_{\varepsilon}$ has a biLipschitz identification with the completion $\bar{Z}$ of $Z$. We also prove that this uniformization procedure can be done at an exponent that is often optimal in the case of CAT$(-1)$ spaces.

Explore related subjects

math.MG

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Clark Butler. 2021-01-03. Uniformizing Gromov hyperbolic spaces with Busemann functions. https://arxiv.org/abs/2007.11143

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

The disjoint disks property for Busemann $G$-spaces

We prove that every finite-dimensional Busemann \(G\)-space of dimension at least five has the disjoint disks property (DDP). For a sufficiently small metric sphere \(L=S(c,r)\), we show that every embedded arc contained in an exact distance level is a homotopical \(Z_2\)-set in \(L\). It follows that \(L\) has the disjoint arc-disk property and the disjoint homotopies property. Daverman's product theorem then gives DDP for \(L\times\mathbb R\), and a local avoidance argument at the center yields DDP for the ambient \(G\)-space. Since finite-dimensional Busemann \(G\)-spaces are generalized manifolds, in dimensions at least five the remaining obstruction to the Busemann conjecture is the resolution problem.

math.MG↗
arXiv2026

Every Compact Metric Space Is Isometrically Embeddable into the Gromov-Hausdorff Space

Let $(\mathcal{M},d_{\mathrm{GH}})$ denote the Gromov-Hausdorff space of isometry classes of nonempty compact metric spaces. We prove that every nonempty compact metric space is isometrically embeddable into $(\mathcal{M},d_{\mathrm{GH}})$. More precisely, for every $D>0$ and every nonempty compact metric space $K$ with $\operatorname{diam} K\le D$, we realize the space of all $1$-Lipschitz functions on $K$ with values in $[0,D]$ as a family of metrics on a fixed Cantor space. Under this realization, the Gromov-Hausdorff distance agrees exactly with the uniform distance between functions, and each resulting metric space has diameter at most $76D$. We also construct finite approximations for which the Gromov-Hausdorff distance is given by an exact formula, together with a uniform approximation estimate.

math.MG↗
arXiv2026

Measure contraction property on isometric leaves and monotone fibres

For finite measures with positive densities on convex Euclidean supports, we prove that $MCP(κ,N)$ passes with unchanged parameters to almost every isometric leaf of an arbitrary nonexpansive map. The proof rests on a sharp contraction inequality for geometric conditional densities, with exponent equal to the leaf codimension. The inherited dimension parameter is optimal. A total-variation limit on resolvent graphs extends the result to inverse fibres of maximal monotone relations, including convex gradient fibres. We also disprove Klartag's curvature-dimension inheritance conjecture by a firmly nonexpansive example in dimension three and a gradient example in dimension four. In codimension one, affinity of the geometric density yields curvature-dimension inheritance. The first example also gives failure on monotone fibres. Both constructions admit arbitrarily large curvature loss, including for a fixed Gaussian ambient measure on families of leaves of positive quotient measure.

math.MG↗