Abstract The objective of this series is to study metric geometric properties of disjoint unions of Cayley graphs of amenable groups by group properties of the Cayley accumulation points in the space of marked groups. In this Part II, we prove that a disjoint union admits a fibred coarse embedding into a Hilbert space (as a disjoint union) if and only if the Cayley boundary of the sequence in the space of marked groups is uniformly a-T-menable. We furthermore extend this result to ones with other target spaces. By combining our main results with constructions of Osajda and Arzhantseva–Osajda, we construct two systems of markings of a certain sequence of finite groups with two opposite extreme behaviors of the resulting two disjoint unions: With respect to one marking, the space has property A. On the other hand, with respect to the other, the space does not admit fibred coarse embeddings into Banach spaces with non-trivial type (for instance, uniformly convex Banach spaces) or Hadamard manifolds; the Cayley limit group is, furthermore, non-exact.

中文翻译：

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Cayley 拓扑和粗几何中的群近似，第二部分：纤维粗嵌入

摘要 本系列的目的是通过标记群空间中凯莱累积点的群性质研究顺从群的凯莱图的不相交并集的度量几何性质。在第二部分中，我们证明了一个不相交的联合允许一个纤维粗嵌入到希尔伯特空间中（作为一个不相交的联合），当且仅当标记群空间中序列的凯莱边界是一致的。我们进一步将此结果扩展到具有其他目标空间的结果。通过将我们的主要结果与 Osajda 和 Arzhantseva-Osajda 的构造相结合，我们构造了有限群的特定序列的两个标记系统，并具有由此产生的两个不相交联合的两个相反的极端行为：对于一个标记，空间具有属性 A . 另一方面，相对于对方，该空间不允许将纤维粗嵌入到具有非平凡类型的 Banach 空间（例如，均匀凸 Banach 空间）或 Hadamard 流形中；此外，凯莱极限群是不精确的。