  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.
