Polynomial growth and asymptotic dimension

Bonamy et al \cite{BBEGLPS} showed that graphs of polynomial growth have finite asymptotic dimension. We refine their result showing that a graph of polynomial growth strictly less than $n^{k+1}$ has asymptotic dimension at most $k$. As a corollary Riemannian manifolds of bounded geometry and polynomial growth strictly less than $n^{k+1}$ have asymptotic dimension at most $k$. We show also that there are graphs of growth $<n^{1+\epsilon}$ for any $\epsilon >0$ and infinite asymptotic Assouad-Nagata dimension.