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
N1 - Funding Information:
The authors wish to express their gratitude to Professor Guoliang Yu and Professor Qin Wang for their kind invitation to Fudan University in Shanghai in July, 2013. Part of this work was done during that stay. Some other part of this work was done during the two-year stay of the first-named author in the École Polytech-nique Fédérale de Lausanne supported by Grant-in-Aid for JSPS Oversea Research Fellowships. The first-named author wishes to express his gratitude to Professor Nicolas Monod and Mrs. Marcia Gouffon for their hospitality and help on his visit. The two authors are grateful to Romain Tessera on LEF groups and for drawing their attention to the example as in Lemma 5.1, and Ana Khukhro and Yves Stalder on wreath products. They also thank Narutaka Ozawa for discussions, and Yuhei Suzuki for the careful reading of the first draft of this paper. The first named author is grateful to Rostislav I. Grigorchuk for providing him with Example 2.8, to Goulnara Arzhantseva for discussions on alternating groups and suggestion for adding Remark 1.7, to Laurent Bartholdi and Anna Erschler for discussions on [9], to Yves de Cornulier and Martin Kassabov respectively for the reference [46, 47, 63], and to Vsevolod Gubarev for the terminology “unimodular” for SL±-group. The authors are grateful to the referee for several comments and for supplying them the references [22, 32].
Funding Information:
This work was in part done while the first-named author was partially supported by JSPS KAKENHI Grant Number JP25800033.
Publisher Copyright:
© 2021 The Author(s).
PY - 2021/3
Y1 - 2021/3
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
UR - http://www.scopus.com/inward/citedby.url?scp=85065838948&partnerID=8YFLogxK
U2 - 10.1142/S1793525320500089
DO - 10.1142/S1793525320500089
M3 - Article
AN - SCOPUS:85065838948
SN - 1793-5253
VL - 13
SP - 1
EP - 47
JO - Journal of Topology and Analysis
JF - Journal of Topology and Analysis
IS - 1
ER -