We introduce a spectrum of monotone coarse invariants for metric measure spaces called Poincaré profiles. The two extremes of this spectrum determine the growth of the space, and the separation profile as defined by Benjamini–Schramm–Timár. In this paper we focus on properties of the Poincaré profiles of groups with polynomial growth, and of hyperbolic spaces, where we deduce a connection between these profiles and conformal dimension. As applications, we use these invariants to show the non-existence of coarse embeddings in a variety of examples.

中文翻译：

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群和空间的庞加莱档案

我们介绍了称为Poincaré轮廓的度量尺度空间的单调粗糙不变量的谱。这个光谱的两个极端决定了空间的增长，以及本杰米尼-施拉姆-提马尔所定义的分离轮廓。在本文中，我们着重于具有多项式增长的群的Poincaré轮廓的性质以及双曲空间的性质，我们在这些性质上推断出这些轮廓与共形维之间的联系。作为应用程序，我们在各种示例中使用这些不变量来显示不存在粗略嵌入。