On asymptotic Assouad-Nagata dimension (Q869681)

A certain analogy between the asymptotic Assouad-Nagata dimension AN-asdim and the asymptotic dimension asdim invites one to transfer the asymptotic dimension theory to the asymptotic Assouad-Nagata dimension. In this paper the authors extend to the Assouad-Nagata dimension the theorem that characterizes the asymptotic dimension of a metric space \(X\) as the covering dimension of the Higson corona \(\nu_L X\) with respect to the sublinear coarse structure on \(X\). Indeed they show that if \(X\) is a cocompact connected proper metric space which has finite asymptotic Assouad-Nagata dimension, then \(\text{AN-as}\!\dim X=\dim\nu_LX\). As an application, they prove the equality \(\text{AN-as}\!\dim(X\times{\mathbb R})= \text{AN-as}\!\dim X+1\) for a cocompact connected proper metric space \(X\).