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]. dy asymptotic dimension with linear control ℓ - asdim ω that depends on a fixed ultrafilter ω on N . It turns out that the asymptotic Assouad–Nagata dimension is the supremum of ℓ - asdim ω over all ω and the asymptotic dimension with linear control is the minimum of ℓ - asdim ω over all ω.