# Groups, Geometry, and Dynamics

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**Volume 5, Issue 4, 2011, pp. 691–728**

**DOI: 10.4171/GGD/145**

Published online: 2011-09-09

Characterizing the Cantor bi-cube in asymptotic categories

Taras Banakh^{[1]}and Ihor Zarichnyi

^{[2]}(1) Ivan Franko National University, Lviv, Ukraine

(2) Ivan Franko National University, Lviv, Ukraine

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.

*Keywords: *Locally finite groups, coarse equivalence, ultrametric spaces, extended Cantor set

Banakh Taras, Zarichnyi Ihor: Characterizing the Cantor bi-cube in asymptotic categories. *Groups Geom. Dyn.* 5 (2011), 691-728. doi: 10.4171/GGD/145