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

Masato Mimura, Hiroki Sako

Research output: Contribution to journalArticlepeer-review

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.

Original languageEnglish
Pages (from-to)62-108
Number of pages47
JournalAnalysis and Geometry in Metric Spaces
Volume7
Issue number1
DOIs
Publication statusPublished - 2019 Jan 1

Keywords

  • Fibred coarse embedding
  • a-T-menability
  • exact groups
  • expanders
  • space of marked groups

ASJC Scopus subject areas

  • Analysis
  • Geometry and Topology
  • Applied Mathematics

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