Group approximation in Cayley topology and coarse geometry, Part I: Coarse embeddings of amenable groups

2013/10/17 by Masato Mimura, Mimura, Masato, Hiroki Sako +1
Mathematics · #Advanced Banach Space Theory #Advanced Operator Algebra Research #Advanced Topics in Algebra #FOS: Mathematics #Group Theory (math.GR) #Metric Geometry (math.MG) #Operator Algebras (math.OA)

paper · pdf · doi:10.48550/arxiv.1310.4736

openalex publication_date 2013/10/17 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

The objective of this series is to study metric geometric properties of (coarse) disjoint unions of amenable Cayley graphs. We employ the Cayley topology and observe connections between large scale structure of metric spaces and group properties of Cayley accumulation points. In this Part I, we prove that a disjoint union has property A of G. Yu if and only if all groups appearing as Cayley accumulation points in the space of marked groups are amenable. As an application, we construct two disjoint unions of finite special linear groups (and unimodular linear groups) with respect to two systems of generators that look similar such that one has property A and the other does not admit (fibred) coarse embeddings into any Banach space with non-trivial type (for instance, any uniformly convex Banach space).

Citations

• The maximal coarse Baum-Connes conjecture for spaces which admit a fibred coarse embedding into Hilbert space

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