Cheeger–Gromoll splitting theorem for groups

Abstract

We study a notion of curvature for finitely generated groups which serves the role of Ricci curvature for Riemannian manifolds. We prove an analog of the Cheeger–Gromoll splitting theorem. As a consequence, we give a geometric characterization of virtually abelian groups. We also explore the relation between this notion of curvature and the growth of groups.