Characterizing the Cantor bi-cube in asymptotic categories

Abstract

We present characterizations of metric spaces that are micro-, macro- or bi-uniformly equivalent to the extended Cantor set EC$=\{{\sum }_{i=-n}^{\infty }\frac{2x_i}{3^i}\mid n\in \mathbb{N},(x_i{)}_{i\in \mathbb{Z}}\in \{0,1{\}}^{\mathbb{Z}}\}\subset \mathbb{R}$, which is bi-uniformly equivalent to the Cantor bi-cube $2^{<\mathbb{Z}}=\{(x_i{)}_{i\in \mathbb{Z}}\in \{0,1{\}}^{\mathbb{Z}}\mid$ there exists $n$ such that $x_i=0$ for all $i\ge n\}$ endowed with the metric $d((x_i),(y_i))=$ max${}_{i\in \mathbb{Z}}{2}^i|x_i-y_i|$. The 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 may be of independent interest.