TY - JOUR

T1 - Group approximation in Cayley topology and coarse geometry Part I

T2 - Coarse embeddings of amenable groups

AU - Mimura, Masato

AU - Sako, Hiroki

PY - 2019/1/1

Y1 - 2019/1/1

N2 - 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 Part I, we prove that a disjoint union has property A of 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 (fibered) coarse embeddings into any Banach space with nontrivial type (for instance, any uniformly convex Banach space).

AB - 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 Part I, we prove that a disjoint union has property A of 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 (fibered) coarse embeddings into any Banach space with nontrivial type (for instance, any uniformly convex Banach space).

KW - Property A

KW - amenability

KW - coarse embeddings

KW - space of marked groups

UR - http://www.scopus.com/inward/record.url?scp=85065838948&partnerID=8YFLogxK

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U2 - 10.1142/S1793525320500089

DO - 10.1142/S1793525320500089

M3 - Article

AN - SCOPUS:85065838948

JO - Journal of Topology and Analysis

JF - Journal of Topology and Analysis

SN - 1793-5253

ER -