JournalsggdVol. 5, No. 4pp. 691–728

Characterizing the Cantor bi-cube in asymptotic categories

  • Taras Banakh

    Ivan Franko National University, Lviv, Ukraine
  • Ihor Zarichnyi

    Ivan Franko National University, Lviv, Ukraine
Characterizing the Cantor bi-cube in asymptotic categories cover
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Abstract

We present characterizations of metric spaces that are micro-, macro- or bi-uniformly equivalent to the extended Cantor set EC={i=n2xi3inN, (xi)iZ{0,1}Z}R= \{\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<Z={(xi)iZ{0,1}Z2^{<\mathbb{Z}}=\{(x_i)_{i\in\mathbb{Z}}\in \{0,1\}^\mathbb{Z}\mid there exists nn such that xi=0x_i=0 for all in}i\ge n\} endowed with the metric d((xi),(yi))=d((x_i),(y_i))= maxiZ2ixiyi_{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.

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