The coarse classification of homogeneous ultra-metric spaces

Abstract

We prove that two homogeneous ultra-metric spaces $X,Y$ are coarsely equivalent if and only if ${\mathrm{Ent}}^{♯}(X)={\mathrm{Ent}}^{♯}(Y)$ where ${\mathrm{Ent}}^{♯}(X)$ 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.

Cited by in corpus (2)

• The coarse classification of countable abelian groups
• Zig-zag chains and metric equivalences between ultrametric spaces