Amenability, locally finite spaces, and bi-lipschitz embeddings

TL;DR

This paper introduces the Small Neighborhood property as a new way to extend amenability to locally finite metric spaces and explores its implications for bi-lipschitz embeddings into Euclidean spaces.

Contribution

It defines the isoperimetric constant for locally finite metric spaces, introduces the SN property, and investigates its relationship with existing notions of amenability and embedding theorems.

Findings

Spaces with uniform bounded geometry and SN property can embed into Euclidean spaces.

Certain metric graphs with SN embed isometrically into Hilbert spaces.

Metric trees without SN do not embed bi-lipschitzly into finite-dimensional Hilbert spaces.

Abstract

We define the isoperimetric constant for any locally finite metric space and we study the property of having isoperimetric constant equal to zero. This property, called Small Neighborhood property, clearly extends amenability to any locally finite space. Therefore, we start making a comparison between this property and other notions of amenability for locally finite metric spaces that have been proposed by Gromov, Lafontaine and Pansu, by Ceccherini-Silberstein, Grigorchuk and de la Harpe and by Block and Weinberger. We discuss possible applications of the property SN in the study of embedding a metric space into another one. In particular, we propose three results: we prove that a certain class of metric graphs that are isometrically embeddable into Hilbert spaces must have the property SN. We also show, by a simple example, that this result is not true replacing property SN with amenability. As a second result, we prove that \emph{many} spaces with \emph{uniform bounded geometry} having a bi-lipschitz embedding into Euclidean spaces must have the property SN. Finally, we prove a Bourgain-like theorem for metric trees: a metric tree with uniform bounded geometry and without property SN does not have bi-lipschitz embeddings into finite-dimensional Hilbert spaces.