Quasi-isometric modification of Gromov-Hausdorff distance
Alexei Naianzin
TL;DR
The paper introduces a quasi‑isometric distance $\hat{d}$ to compare arbitrary quasi‑isometric spaces and proves a meaningful hierarchy with Gromov–Hausdorff convergence, namely GH $\Rightarrow$ QI $\Rightarrow$ PGH. It shows that $\hat{d}$ is not a metric but is closely linked to a genuine metric $D$ that induces the same topology and coarse structure on the space of metric spaces, and establishes quantitative bounds $\hat{d}(X,Y)=r \Rightarrow D(X,Y)\le \ln(1+2r)$ and $D(X,Y)=r \Rightarrow \hat{d}(X,Y)\le e^{2r}-e^{r}$. The authors prove that qi convergence preserves a range of metric properties (e.g., total boundedness, separability, properness under completeness, intrinsicity, geodesicity, $\delta$-hyperbolicity, and $\mathrm{CAT}^\kappa$) and show that the space of all metric spaces equipped with $D$ is path‑connected via explicit continuous deformations whose length is bounded by a function of the qi constant. They also develop a coarse geometry framework, showing that the associated coarse structure on equivalence classes is monogenic and metrizable, with a countable generating family, and provide an explicit generalized metric $\rho$ that preserves the coarse structure. Overall, the work offers a robust, coarse‑mensitive extension of GH theory that enables comparing noncompact and structurally diverse spaces through deformation‑theoretic methods.
Abstract
We define a distance analogous to the Gromov-Hausdorff distance that enables the comparison of arbitrary quasi-isometric spaces. We also investigate properties preserved under limits with respect to this distance, as well as properties of the entire class of metric spaces equipped with this distance. For this purpose, we introduce the notion of quasi-isometric distortion for correspondences. Using this notion, we prove that the class of all metric spaces is path-connected; in fact, any two metric spaces can be connected by a curve of finite length.