Conjugation curvature for Cayley graphs

TL;DR

This paper introduces a finite-radius Ricci curvature concept for Cayley graphs, linking geometric properties with algebraic group features, and demonstrating its effectiveness across various group types.

Contribution

It defines conjugation curvature for Cayley graphs, connecting geometric group theory with algebraic properties and analyzing its behavior across different groups.

Findings

Abelian groups are flat under conjugation curvature.

Zero curvature implies the group is virtually abelian.

Captures curvature phenomena in right-angled Artin and nilpotent groups.

Abstract

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.