We present characterizations of metric spaces that are micro-, macro- or bi-uniformly equivalent to the extended Cantor set EC={∑i=−n∞3i2xi∣n∈N, (xi)i∈Z∈{0,1}Z}⊂R, which is bi-uniformly equivalent to the Cantor bi-cube 2<Z={(xi)i∈Z∈{0,1}Z∣ there exists n such that xi=0 for all i≥n} endowed with the metric d((xi),(yi))= maxi∈Z2i∣xi−yi∣. 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.
Classification :
20-XX, 00-XX
Mots-clés :
Locally finite groups, coarse equivalence, ultrametric spaces, extended Cantor set
Affiliations des auteurs :
Taras Banakh
1
;
Ihor Zarichnyi
1
1
Ivan Franko National University, Lviv, Ukraine
Taras Banakh; Ihor Zarichnyi. Characterizing the Cantor bi-cube in asymptotic categories. Groups, geometry, and dynamics, Tome 5 (2011) no. 4, pp. 691-728. doi: 10.4171/ggd/145
@article{10_4171_ggd_145,
author = {Taras Banakh and Ihor Zarichnyi},
title = {Characterizing the {Cantor} bi-cube in asymptotic categories},
journal = {Groups, geometry, and dynamics},
pages = {691--728},
year = {2011},
volume = {5},
number = {4},
doi = {10.4171/ggd/145},
url = {http://geodesic.mathdoc.fr/articles/10.4171/ggd/145/}
}
TY - JOUR
AU - Taras Banakh
AU - Ihor Zarichnyi
TI - Characterizing the Cantor bi-cube in asymptotic categories
JO - Groups, geometry, and dynamics
PY - 2011
SP - 691
EP - 728
VL - 5
IS - 4
UR - http://geodesic.mathdoc.fr/articles/10.4171/ggd/145/
DO - 10.4171/ggd/145
ID - 10_4171_ggd_145
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%J Groups, geometry, and dynamics
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%P 691-728
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%N 4
%U http://geodesic.mathdoc.fr/articles/10.4171/ggd/145/
%R 10.4171/ggd/145
%F 10_4171_ggd_145
