Characterizing the Cantor bi-cube in asymptotic categories

2009/08/31 by Taras Banakh, Ihor Zarichnyi · 1 citation
Mathematics · #math.MG #math.GR #math.GT #msc:54E35 #msc:54E40

paper · pdf

published as Groups, Geometry, and Dynamics, Vol.5, No.4 (2011) 691-728 · 24 pages; the paper now contains three characterization theorems for the extended Cantor set, which resolves all open problems posed in the preceding version of the paper

arxiv created 2011/10/10 · arxiv updated 2011/10/11

Abstract

We present the characterization of metric spaces that are micro-, macro- or bi-uniformly equivalent to the extended Cantor set \∑_i=-n^∞(2x_i)/(3ⁱ):n∈\IN , (x_i)_i∈\IZ∈\0,1\^\IZ\⊂\IR, which is bi-uniformly equivalent to the Cantor bi-cube 2^<\IZ=\(x_i)_i∈\IZ∈ \0,1\^\IZ:∃ n ∀ i≥ n x_i=0\ endowed with the metric d((x_i),(y_i))=max_i∈\IZ2ⁱ|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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• Classifying homogeneous ultrametric spaces up to coarse equivalence

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