Ollivier--Ricci Curvature on Groups of Polynomial Growth

Camillo Brena, Elia Bruè

arXiv:2608.14232v1 Primary: math.DG Secondary: math.MG, math.PR

Abstract

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 $k\in\mathbb{N}$.

Disclosure

“e proof of Theorem 3.5, specifically in verifying that the claimed set of generators can be chosen as stated. Finally, ChatGPT suggested the observation in Remark 3.4 after the authors prompted it with the group to look for the example. ChatGPT was also used for proofreading. All mathematical arguments, computations, and conclusions were independently verified by the authors. No text in this article was written by AI. 1.7. Acknowledgments. Part of this work was carried out whil”
PDF page 9

Classification: Proof ideas or individual proof-step assistance
Multiplier: 8
Verified: 2026-08-23T05:14:57.902520Z

Structural counts

Pages 36 pdf

Theorems 9 source

Lemmas 9 source

Propositions 4 source

Corollaries 0 source

Definitions 4 source

Displayed equations 242 source

Bibliography entries 41 source

Appendix pages 0 estimated

Count notes

Source counts use the expanded primary TeX file paper_august_2026.tex.
Appendix pages include the first PDF page with an explicit Appendix heading through the final page.