  We introduce a notion of Ricci curvature for Cayley graphs that can be
thought of as "medium-scale" because it is neither infinitesimal nor
asymptotic, but based on a chosen finite radius parameter. We argue that it
gives the foundation for a definition of Ricci curvature well adapted to
geometric group theory, beginning by observing that the sign can easily be
characterized in terms of conjugation in the group. With this conjugation
curvature $\kappa$, abelian groups are identically flat, and in the other
direction we show that $\kappa\equiv 0$ implies the group is virtually abelian.
Beyond that, $\kappa$ captures known curvature phenomena in right-angled Artin
groups (including free groups) and nilpotent groups, and has a strong
relationship to other group-theoretic notions like growth rate and dead ends.
We study dependence on generators and behavior under embeddings, and close with
directions for further development and study.
