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) (Formula presented.) for (Formula presented.), or (Formula presented.) whenever (Formula presented.) is a bounded degree graph with subexponential growth, where (Formula presented.) 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 (Formula presented.) whenever (Formula presented.) is a bounded degree graph with at most polynomial growth, and no quasi-isometric embedding whenever (Formula presented.) has subexponential growth. Finally, we show that there is no regular map (Formula presented.) where (Formula presented.) 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.