The coarse classification of homogeneous ultra-metric spaces

Details The coarse classification of homogeneous ultra-metric spaces The coarse classification of homogeneous ultra-metric spaces

2008-01-14

arxiv.org/abs/0801.2132v1

Mathematics

Geometric Topology

Scientific paper

We prove that two homogeneous ultra-metric spaces $X,Y$ are coarsely equivalent if and only if $\mathrm{Ent}^\sharp(X)=\mathrm{Ent}^\sharp(Y)$ where $\mathrm{Ent}^\sharp(X)$ is the so-called sharp entropy of $X$. This classification implies that each homogeneous proper ultra-metric space is coarsely equivalent to the anti-Cantor set $2^{<\omega}$. For the proof of these results we develop a technique of towers which can have an independent interest.

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