# 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$={∑_{i=−n}3_{i}2x_{i} ∣n∈N,(x_{i})_{i∈Z}∈{0,1}_{Z}}⊂R$, which is bi-uniformly equivalent to the Cantor bi-cube $2_{<Z}={(x_{i})_{i∈Z}∈{0,1}_{Z}∣$ there exists $n$ such that $x_{i}=0$ for all $i≥n}$ endowed with the metric $d((x_{i}),(y_{i}))=$ max$_{i∈Z}2_{i}∣x_{i}−y_{i}∣$. 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