On strong embeddability and finite decomposition complexity

The strong embeddability is a notion of metric geometry, which is an intermediate property lying between coarse embeddability and property A. In this paper, we study the permanence properties of strong embeddability for metric spaces. We show that strong embeddability is coarsely invariant and it is closed under taking subspaces, direct products, direct limits and finite unions. Furthermore, we show that a metric space is strongly embeddable if and only if it has weak finite decomposition complexity with respect to strong embeddability.