Characterizing the Cantor bi-cube in asymptotic categories

TL;DR

This paper characterizes metric spaces equivalent to the Cantor bi-cube in various uniform categories, revealing universal properties among ultrametric spaces and countable groups, and introduces a novel tower technique.

Contribution

It provides new characterizations of spaces bi-uniformly equivalent to the Cantor bi-cube and develops a tower method with potential broader applications.

Findings

Any two uncountable proper isometrically homogeneous ultrametric spaces are coarsely equivalent.

Any two countable locally finite groups with proper left-invariant metrics are coarsely equivalent.

The tower technique introduced may have independent interest.

Abstract

We present the characterization of metric spaces that are micro-, macro- or bi-uniformly equivalent to the extended Cantor set $\{\sum_{i=-n}^\infty\frac{2x_i}{3^i}:n\in\IN ,\;(x_i)_{i\in\IZ}\in\{0,1\}^\IZ\}\subset\IR$, which is bi-uniformly equivalent to the Cantor bi-cube $2^{<\IZ}=\{(x_i)_{i\in\IZ}\in \{0,1\}^\IZ:\exists n\;\forall i\ge n\;x_i=0\}$ endowed with the metric $d((x_i),(y_i))=\max_{i\in\IZ}2^i|x_i-y_i|$. Those characterizations imply that any two (uncountable) proper isometrically homogeneous ultrametric spaces are coarsely (and bi-uniformly) equivalent. This implies that any two countable locally finite groups endowed with proper left-invariant metrics are coarsely equivalent. For the proof of these results we develop a technique of towers which can have an independent interest.