Strong Embeddability Of Discrete Metric Spaces

In 2013, Ji,Ogle and Ramsey [28] introduced a new notion,strong embeddability for metric spaces,it is an intermediate property of metric geometry lying between coarse embeddability and property A, and it is closed under arbitrary group extensions.In [46],Guoliang Yu proved that the coarse Baum-Connes conjecture holds for the metric spaces with bounded geometry which admit a coarse embedding into Hilbert space. Therefore, if a metric space with bounded geometry is strongly embeddable,then the coarse Baum-Connes conjecture holds for the space naturally.For a new property of metric spaces in coarse geometry,we generally study the permanence properties under standard constructions so that we can construct more metric spaces which possess this property.In view of this, it is natural to ask which standard constructions can preserve the strong embeddability for metric spaces,thus,we can construct more metric spaces which satisfy the coarse Baum-Connes conjecture.In this paper,we study permanence properties of strong embeddability for discrete metric spaces under operations in coarse geometry.We prove that strong embeddability is coarsely invariant; we show that strong embeddability is closed under taking subspaces,direct products,direct limits,finite unions and certain infinite unions;we prove that strong embeddability is preserved under relatively hyperbolic groups,free products and amalgamated free products,the fundamental groups of graph of groups and finite decomposition complexity.