Ollivier--Ricci Curvature on Groups of Polynomial Growth

We study Ollivier–Ricci curvature on Cayley graphs of groups of polynomial growth. Our main result shows that non-negative Ollivier–Ricci curvature forces the group to be virtually abelian. As an application, we prove that connected vertex-transitive graphs of polynomial growth and non-negative Ollivier–Ricci curvature are quasi-isometric to ${\mathbb{Z}}^k$, for some $k\in \mathbb{N}$.