  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.
