We present characterizations of metric spaces that are micro-, macro- or bi-uniformly equivalent to the extended Cantor set EC, which is bi-uniformly equivalent to the Cantor bi-cube there exists such that for all endowed with the metric max. 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. 691–728DOI 10.4171/GGD/145