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.
Categories: math.MG math.CO