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 [4].We study asymptotic dimension with linear control ℓ-asdimωℓ-asdimω that depends on a fixed ultrafilter ω   on NN. It turns out that the asymptotic Assouad–Nagata dimension is the supremum of ℓ-asdimωℓ-asdimω over all ω   and the asymptotic dimension with linear control is the minimum of ℓ-asdimωℓ-asdimω over all ω.