Medium-Scale Ricci Curvature for Hyperbolic Groups

We study the relationship between a notion of medium-scale Ricci curvature for finitely generated groups and that of hyperbolicity in the sense of Gromov. We give an example of a generating set that gives zero curvature with positive density for the free group of rank 2. We prove that, by making the radius used in computing the curvature sufficiently large, we can always have negative curvature outside of a ball in non-elementary hyperbolic groups. On the other hand, we give an example of a group which has negative curvature for all non-identity points but is not hyperbolic.