TY - JOUR

T1 - Group Approximation in Cayley Topology and Coarse Geometry, Part II

T2 - Fibred Coarse Embeddings

AU - Mimura, Masato

AU - Sako, Hiroki

N1 - Publisher Copyright:
© 2019 Masato Mimura et al., published by De Gruyter Open.

PY - 2019/1/1

Y1 - 2019/1/1

N2 - 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.

AB - 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.

KW - Fibred coarse embedding

KW - a-T-menability

KW - exact groups

KW - expanders

KW - space of marked groups

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U2 - 10.1515/agms-2019-0005

DO - 10.1515/agms-2019-0005

M3 - Article

AN - SCOPUS:85073813275

VL - 7

SP - 62

EP - 108

JO - Analysis and Geometry in Metric Spaces

JF - Analysis and Geometry in Metric Spaces

SN - 2299-3274

IS - 1

ER -