The objective of this series is to study metric geometric properties of (coarse) disjoint unions of amenable Cayley graphs. We employ the Cayley topology and observe connections between large scale structure of metric spaces and group properties of Cayley accumulation points. In Part I, we prove that a disjoint union has property A of Yu if and only if all groups appearing as Cayley accumulation points in the space of marked groups are amenable. As an application, we construct two disjoint unions of finite special linear groups (and unimodular linear groups) with respect to two systems of generators that look similar such that one has property A and the other does not admit (fibered) coarse embeddings into any Banach space with nontrivial type (for instance, any uniformly convex Banach space).

中文翻译：

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Cayley 拓扑和粗几何中的群逼近第一部分：顺从群的粗嵌入

本系列的目的是研究可服从凯莱图的（粗）不相交并集的度量几何特性。我们采用 Cayley 拓扑并观察度量空间的大尺度结构与 Cayley 累积点的群性质之间的联系。在第一部分中，我们证明了一个不相交的并集具有 Yu 的性质 A 当且仅当所有在标记群空间中作为 Cayley 累积点出现的群都是服从的。作为一个应用程序，我们针对两个看起来相似的生成器系统构造了两个不相交的有限特殊线性群（和单模线性群）并集，其中一个具有属性 A，另一个不允许（纤维化）粗略嵌入到任何 Banach具有非平凡类型的空间（例如，任何均匀凸的 Banach 空间）。