Asymptotic dimension for covers with controlled growth

We prove various obstructions to the existence of regular maps (or coarse embeddings) between commonly studied spaces. For instance, there is no regular map (or coarse embedding) $H^n\to H^{n-1}\times Y$ for $n⩾3$, or ${\left(T_3\right)}^n\to {\left(T_3\right)}^{n-1}\times Y$ whenever $Y$ is a bounded degree graph with subexponential growth, where $T_3$ is the 3-regular tree. We also resolve Question 5.2 (Groups Geom. Dyn. 6 (2012), no. 4, 639–658), proving that there is no regular map $H^2\to T_3\times Y$ whenever $Y$ is a bounded degree graph with at most polynomial growth, and no quasi-isometric embedding whenever $Y$ has subexponential growth. Finally, we show that there is no regular map $F^n\to Z\wr F^{n-1}$ where $F$ is the free group on two generators. To prove these results, we introduce and study generalisations of asymptotic dimension that allow unbounded covers with controlled growth.