Strong embeddability and extensions of groups

TL;DR

The paper introduces strong embeddability for metric spaces, explores its properties, and shows it is preserved under group extensions, bridging coarse embeddability and property A.

Contribution

It defines strong embeddability, develops a relative version, and proves its preservation under group extensions, advancing understanding of metric space embeddings.

Findings

Strong embeddability lies between coarse embeddability and property A.

A relative version of strong embeddability is formulated for metric spaces.

Strong embeddability is preserved under group extensions.

Abstract

We introduce the notion of strong embeddability for a metric space. This property lies between coarse embeddability and property A. A relative version of strong embeddability is developed in terms of a family of set maps on the metric space. When restricted to discrete groups, this yields relative coarse embeddability. We verify that groups acting on a metric space which is strongly embeddable has this relative strong embeddability, provided the stabilizer subgroups do. As a corollary, strong embeddability is preserved under group extensions.