  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) $\mathbb H^n\to\mathbb H^{n-1}\times Y$ for $n\geq
3$, or $(T_3)^n \to (T_3)^{n-1}\times Y$ whenever $Y$ is a bounded degree graph
with subexponential growth, where $T_3$ is the $3$-regular tree.
  We also resolve a question of Benjamini-Schramm-Tim\'ar, proving that there
is no regular map $\mathbb H^2 \to T_3 \times Y$ 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 $F^n\to \mathbb Z\wr F^{n-1}$
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).
