Abstract. We study property A defined by G. Yu and the operator norm localization property defined by X. Chen, R. Tessera, X. Wang, and G. Yu. These are coarse geometric properties for metric spaces, which have applications to operator K-theory. It is proved that the two properties are equivalent for discrete metric spaces with bounded geometry.
We get three types of results on measurable group theory; direct product groups of Ozawa's class S groups, wreath product groups and amalgamated free products. We prove measure equivalence factorization results on direct product groups of Ozawa's class S groups. As consequences, Monod-Shalom type orbit equivalence rigidity theorems follow. We prove that if two wreath product groups A G, B Γ of nonamenable exact direct product groups G, Γ with amenable bases A, B are measure equivalent, then G and Γ are measure equivalent. We get Bass-Serre rigidity results on amalgamated free products of nonamenable exact direct product groups.
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.
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).
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.