Medium-Scale Ricci Curvature for Hyperbolic Groups

TL;DR

This paper explores the concept of medium-scale Ricci curvature in finitely generated groups and its connection to hyperbolicity, providing examples and conditions under which curvature indicates hyperbolic behavior.

Contribution

It introduces a notion of medium-scale Ricci curvature for groups, analyzes its relation to hyperbolicity, and presents examples illustrating the nuanced behavior of curvature in different groups.

Findings

Zero curvature with positive density in free groups.

Negative curvature outside a ball in hyperbolic groups.

Existence of non-hyperbolic groups with negative curvature everywhere.

Abstract

We study the relationship between a notion of medium-scale Ricci curvature for finitely generated groups and that of hyperbolicity in the sense of Gromov. We give an example of a generating set that gives zero curvature with positive density for the free group of rank 2. We prove that, by making the radius used in computing the curvature sufficiently large, we can always have negative curvature outside of a ball in non-elementary hyperbolic groups. On the other hand, we give an example of a group which has negative curvature for all non-identity points but is not hyperbolic.