On Asymptotic Dimension With Linear Control

We construct a countable p-local group with a proper invariant metric whose Assouad-Nagata dimension is strictly greater than the asymptotic dimension with linear control. This solves Problem 8.6 from the list [Dr]. We study asymptotic dimension with linear control $\ell$-$asdim_{\omega}$ that depends on a fixed ultrafilter on the naturals. It turns out that the Assouad-Nagata dimension is the supremum of $\ell$-$asdim_{\omega}$ on all $\omega$ and the asymptotic dimension with linear control is the minimum.