Medium-scale curvature for Cayley graphs

Assaf Bar-Natan, Moon Duchin, Robert Kropholler

Published 2017-12-07Version 1

We introduce a notion of Ricci curvature for Cayley graphs that can be called medium-scale because it is neither infinitesimal nor asymptotic, but based on a chosen finite radius parameter. For this definition, abelian groups are identically flat, and on the other hand we show that $\kappa\equiv 0$ implies the group is virtually abelian. In right-angled Artin groups, the curvature is zero on central elements and negative otherwise. On the other hand, we find nilpotent, CAT(0), and hyperbolic groups with points of positive curvature. We study dependence on generators and behavior under embeddings.