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arXiv · 0908.3687

Characterizing the Cantor bi-cube in asymptotic categories

Taras Banakh·Ihor Zarichnyi
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Abstract

We present the characterization of metric spaces that are micro-, macro- or bi-uniformly equivalent to the extended Cantor set $\{\sum_{i=-n}^\infty\frac{2x_i}{3^i}:n\in\IN ,\;(x_i)_{i\in\IZ}\in\{0,1\}^\IZ\}\subset\IR$, which is bi-uniformly equivalent to the Cantor bi-cube $2^{<\IZ}=\{(x_i)_{i\in\IZ}\in \{0,1\}^\IZ:\exists n\;\forall i\ge n\;x_i=0\}$ endowed with the metric $d((x_i),(y_i))=\max_{i\in\IZ}2^i|x_i-y_i|$. Those 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 can have an independent interest.

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Taras Banakh, Ihor Zarichnyi. 2009-08-25. Characterizing the Cantor bi-cube in asymptotic categories. https://arxiv.org/abs/0908.3687

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