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) Hn→Hn−1×Y for n≥3, or (T3)n→(T3)n−1×Y whenever Y is a bounded degree graph with subexponential growth, where T3 is the 3-regular tree.
We also resolve a question of Benjamini-Schramm-Timár, proving that there is no regular map H2→T3×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 Fn→Z≀Fn−1 where F is the free group on two generators.
To prove these results, we introduce and study generalizations of asymptotic dimension which allow unbounded covers with controlled growth. For bounded degree graphs, these invariants are monotone with respect to regular maps (hence coarse embeddings).