A Hurewicz‐type theorem for the dynamic asymptotic dimension with applications to coarse geometry and dynamics

We prove a Hurewicz‐type theorem for the dynamic asymptotic dimension originally introduced by Guentner, Willett, and Yu. Calculations of (or simply upper bounds on) this dimension are known to have implications related to cohomology of group actions and the ‐theory of their transformation group ‐algebras. Moreover, these implications are relevant to the current classification program for ‐algebras. As a corollary of our main theorem, we show that the dynamic asymptotic dimension of actions by groups on profinite completions along sequential filtrations by normal subgroups is often subadditive over extensions of groups, which shows that many such actions by elementary amenable groups are finite dimensional. We combine this extension theorem with other novel results relating the dynamic asymptotic dimension of such actions to the asymptotic dimension of corresponding box spaces. This allows us to give upper bounds on the asymptotic dimension of many box spaces (including those of infinitely many groups with exponential growth). For some of these examples, we can also find lower bounds by utilizing the theory of ends of groups.