Remarks on strong embeddability for discrete metric spaces and groups

TL;DR

This paper investigates properties of strong embeddability in discrete metric spaces and groups, demonstrating its stability under certain operations and establishing conditions under which groups inherit strong embeddability.

Contribution

It proves that strong embeddability is preserved under fibering and direct limits, and shows that finitely generated groups with specific actions are strongly embeddable.

Findings

Strong embeddability has fibering permanence property.

Strong embeddability is preserved under direct limits.

Finitely generated groups with coarse quasi-actions on spaces with finite asymptotic dimension are strongly embeddable.

Abstract

In this paper, we show that the strong embeddability has fibering permanence property and is preserved under the direct limit for the metric space. Moreover, we show the following result: let $G$ is a finitely generated group with a coarse quasi-action on a metric space $X$. If $X$ has finite asymptotic dimension and the quasi-stabilizers are strongly embeddable, then $G$ is also strongly embeddable.