Group Approximation in Cayley Topology and Coarse Geometry, Part II: Fibred Coarse Embeddings
Group Approximation in Cayley Topology and Coarse Geometry, Part II: Fibred Coarse Embeddings
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DOI：
10.1515/agms-2019-0005
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发表时间：
2018-04
影响因子：
1
通讯作者：
M. Mimura;Hiroki Sako
中科院分区：
文献类型：
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作者：
M. Mimura;Hiroki Sako
本文的目的是利用标记群空间中Cayley聚点的群性质，研究顺从群Cayley图的不交并的度量几何性质。在第二部分中，我们证明了一个不相交的并允许一个不相交的粗嵌入到一个Hilbert空间中（作为一个不相交的并）当且仅当序列在标记群空间中的Cayley边界是一致a-T-可表的。我们进一步将这个结果推广到其他目标空间。通过将我们的主要结果与Osajda和Arzhantseva-Osajda的构造相结合，我们构造了有限群序列的两个标记系统，其结果的两个不相交并具有两个相反的极端行为：对于一个标记，空间具有性质A.另一方面，相对于另一个，空间不允许嵌入到具有非平凡类型的Banach空间（例如，一致凸Banach空间）或Hadamard流形中; Cayley极限群是非精确的。
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.