Recently I have been studying amenability of groups and property A, and I came across the Higson-Roe condition:
Let $X$ be a uniformly discrete metric space with bounded geometry. $X$ has property $A$ if and only if for every $\varepsilon>0$ and every $R>0$ there exists a map $\xi: X \rightarrow \ell_1(X)_{1,+}, x \mapsto \xi_x$, and a number $S>0$ such that $\left\|\xi_x-\xi_y\right\| \leq \varepsilon$ if $d(x, y) \leq R$ and $\text{supp }\xi_x \subseteq B(x, S)$.
To put things in context, Property A can be thought of as a non-equivariant version of amenability, and is a sufficient condition for (coarse) Hilbert space embeddability. I reproduce the definition below (for more details, see this paper):
Let $X$ be a uniformly discrete metric space. We say that $X$ has property $A$ if for every $\varepsilon>0$ and $R>0$ there exists a collection of finite subsets $\left\{A_x\right\}_{x \in X}, A_x \subseteq X \times \mathbb{N}$ for every $x \in X$, and a constant $S>0$ such that $\frac{\#\left(A_x \triangle A_y\right)}{\#\left(A_x \cap A_y\right)} \leq \varepsilon$ when $d(x, y) \leq R$, and $A_x \subseteq B(x, S) \times \mathbb{N}$.
I studied the equivalence of Higson-Roe condition with property A on uniformly discrete metric spaces with bounded geometry (see this paper). I am interested in knowing more applications of the Higson-Roe condition, but being a novice I don't know where to start. Can anyone give an insight on this? This is mainly a reference request.
