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BY 4.0 license Open Access Published by De Gruyter Open Access August 19, 2019

Group Approximation in Cayley Topology and Coarse Geometry, Part II: Fibred Coarse Embeddings

  • Masato Mimura EMAIL logo and Hiroki Sako


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.


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Received: 2018-05-17
Revised: 2019-03-14
Accepted: 2019-06-04
Published Online: 2019-08-19

© 2019 Masato Mimura et al., published by De Gruyter Open

This work is licensed under the Creative Commons Attribution 4.0 Public License.

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