Appears in collection : Geometry in non-positive curvature and Kähler groups

Asymptotic dimension and n-Uryson width are useful notions of dimension in coarse and systolic geometry respectively. I will explain how using similar techniques one obtains: 1. Sharp estimates for the asymptotic dimension of graphs of polynomial growth 2. A new proof of a theorem of Guth on the n-Uryson width of Riemannian manifolds of small volume growth. This leads to new proof of Gromov's systolic inequality.