Expansion properties of metric spaces not admitting a coarse embedding into a Hilbert space

The main purpose of the paper is to find some expansion properties of locally finite metric spaces which do not embed coarsely into a Hilbert space. The obtained result is used to show that infinite locally finite graphs excluding a minor embed coarsely into a Hilbert space. In an appendix a direct proof of the latter result is given.

💡 Research Summary

The paper investigates the geometric and combinatorial features of locally finite metric spaces that cannot be coarsely embedded into a Hilbert space. A coarse (or “large‑scale”) embedding of a metric space (X,d_X) into another metric space (Y,d_Y) is a map f:X→Y together with two non‑decreasing control functions ρ₁,ρ₂: