Property A and asymptotic dimension

Abstract

The purpose of this note is to characterize the asymptotic dimension $asdim(X)$ of metric spaces $X$ in terms similar to Property A of Yu: If $(X,d)$ is a metric space and $n\ge 0$, then the following conditions are equivalent: [a.] $asdim(X,d)\le n$, [b.] For each $R,\epsilon >0$ there is $S>0$ and finite non-empty subsets $A_x\subset B(x,S)\times N$, $x\in X$, such that $\frac{|A_x\Delta A_y|}{|A_x\cap A_y|}<\epsilon$ if $d(x,y)<R$ and the projection of $A_x$ onto $X$ contains at most $n+1$ elements for all $x\in X$, [c.] For each $R>0$ there is $S>0$ and finite non-empty subsets $A_x\subset B(x,S)\times N$, $x\in X$, such that $\frac{|A_x\Delta A_y|}{|A_x\cap A_y|}<\frac{1}{n+1}$ if $d(x,y)<R$ and the projection of $A_x$ onto $X$ contains at most $n+1$ elements for all $x\in X$.

3 pages

References in corpus (1)

• Some notes on property A

Cited by in corpus (2)

• On equivariant asymptotic dimension
• Decomposition complexity with respect to coarse properties