Polynomial growth and asymptotic dimension

Bonamy et al. [4] 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.