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The objective of this series is to study metric geometric properties ofdisjoint unions of amenable Cayley graphs by group properties of the Cayleyaccumulation points in the space of marked groups. In this Part II, we provethat a disjoint union admits a fibred coarse embedding into a Hilbert space (ina generalized sense) if and only if the Cayley boundary of the sequence in thespace of marked groups is uniformly a-T-menable. We furthermore extend thisresult to ones with other target spaces. By combining our main results withconstructions of Arzhantseva--Osajda and Osajda, we construct two systems ofmarkings of a certain sequence of finite groups with two opposite extremebehaviors 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, thespace does not admit fibred coarse embeddings into Banach spaces withnon-trivial type (for instance, uniformly convex Banach spaces) or Hadamardmanifolds; the Cayley limit group is, furthermore, non-exact.

Group approximation in Cayley topology and coarse geometry, Part II: Fibered coarse embeddings