Growth of the asymptotic dimension function for groups

It is relatively easy to construct a finitely generated group with infinite asymptotic dimension: the restricted wreath product of $\mathbb{Z}$ by $\mathbb{Z}$ provides an example. In light of this, it becomes interesting to consider the rate of growth of the asymptotic dimension function of a group. Loosely speaking, we measure the dimension on $\lambda$-scale and let $\lambda$ increase to infinity to recover the asymptotic dimension. In this paper we consider how the asymptotic dimension function is affected by different constructions involving groups.