# Characterizing the Cantor bi-cube in asymptotic categories

### Taras Banakh

Ivan Franko National University, Lviv, Ukraine### Ihor Zarichnyi

Ivan Franko National University, Lviv, Ukraine

## 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.

## Cite this article

Taras Banakh, Ihor Zarichnyi, Characterizing the Cantor bi-cube in asymptotic categories. Groups Geom. Dyn. 5 (2011), no. 4, pp. 691–728

DOI 10.4171/GGD/145