标题： 可约性，局部有限空间和双利普希茨嵌入 显示英文标题

标题： Amenability, locally finite spaces, and bi-lipschitz embeddings

摘要： 我们定义了任何局部有限度量空间的等周常数，并研究了等周常数等于零的性质。 这一性质被称为小邻域性质，显然将可约性扩展到了任何局部有限空间。 因此，我们开始将这一性质与Gromov、Lafontaine和Pansu、Ceccherini-Silberstein、Grigorchuk和de la Harpe以及Block和Weinberger提出的局部有限度量空间的其他可约性概念进行比较。 我们讨论了小邻域性质在研究一个度量空间嵌入到另一个度量空间中的可能应用。 特别是，我们提出了三个结果：我们证明了一类可以等距嵌入到希尔伯特空间中的度量图必须具有小邻域性质。 我们还通过一个简单的例子表明，如果将小邻域性质替换为可约性，这个结果并不成立。 作为第二个结果，我们证明了具有\emph{许多}结构且\emph{均匀有界几何}具有双利普希茨嵌入到欧几里得空间中的空间必须具有小邻域性质。 最后，我们证明了一个类似Bourgain的定理：具有均匀有界几何且不具有小邻域性质的度量树不能双利普希茨嵌入到有限维希尔伯特空间中。 显示英文摘要

摘要： 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.