Remarks on coarse triviality of asymptotic Assouad-Nagata dimension

TL;DR

This paper demonstrates that for any metric space with a given asymptotic dimension, one can find a hyperbolic metric equivalent in a coarse and topological sense that preserves the asymptotic Assouad-Nagata dimension, and provides counterexamples to certain inequalities.

Contribution

It introduces a method to construct hyperbolic metrics preserving asymptotic Assouad-Nagata dimension and presents counterexamples to the logarithmic law for AN-asdim.

Findings

Existence of hyperbolic metrics preserving asymptotic Assouad-Nagata dimension.

Counterexamples to the logarithmic law for AN-asdim.

A potential counterexample to a Morita-type theorem for AN-asdim.

Abstract

We show for a given metric space $(X,d)$ of asymptotic dimension $n$ that there exists a coarsely and topologically equivalent hyperbolic metric $d'$ of the form $d' = f \circ d$ such that $(X,d')$ is of asymptotic Assouad-Nagata dimension $n$. As a corollary we construct examples of spaces realising strict inequality in the logarithmic law for AN-asdim of a Cartesian product. One of them may be viewed as a counterexample to a specific kind of a Morita-type theorem for AN-asdim.