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

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).

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

“…Space of k-marked groups and Cayley topology. We recall basic facts of the Cayley topology from our Part I paper [MS13]; see Subsection 2.1 there for more details. Fix k ∈ N ≥1 .…”

“…where m denotes a coarse disjoint union (see [MS13,Definition 2.17.(2)] and Subsection 3.2). Chen-Wang-Wang [CWW13] showed that G admits a fibred coarse embedding into a Hilbert space if and only if G is a-T-menable.…”

Amenability versus non‐exactness of dense subgroups of a compact group

“…Here, we briefly recall the definition of it and diagonal products associated to LEF approximations. We refer the reader to [, Subsections 2.1, 2.3, and 5.1] for more details and references. Throughout this paper, for a group , write its unit as .…”

“…In this convergence, we disregard behaviors of outside the ‐ball , where ; this description will get clearer after the reader consults Remark . We refer the reader to [, Lemma 5.1] and the proof of it as another pedagogical example.…”

“…The argument above explains why the proof of Lemma does not imply that the standard wreath product of two finitely generated RF groups is RF. In fact, Gruenberg showed that if is RF for an infinite , then must be abelian ; see [, Subsection 2.2 ] for more details and references.…”