Transitive graphs of non-negative Ollivier-Ricci curvature have polynomial growth

Camillo Brena; Tom Hutchcroft; Florentin Münch

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arXiv · 2512.03968

Transitive graphs of non-negative Ollivier-Ricci curvature have polynomial growth

Camillo Brena·Tom Hutchcroft·Florentin Münch

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Abstract

We prove that every transitive graph of non-negative Ollivier-Ricci curvature has polynomial growth. We deduce from this and a theorem of Brena and Brue that a finitely generated group admits a Cayley graph of non-negative Ollivier-Ricci curvature if and only if it is virtually abelian. Beyond the transitive setting, we also prove a quasi-polynomial $r^{O(\sqrt{\log r})}$ volume bound for balls around typical vertices in finite bounded-degree graphs of non-negative Ollivier-Ricci curvature, greatly strengthening a theorem of Salez (GAFA 2022).

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BibTeX RIS

Camillo Brena, Tom Hutchcroft, Florentin Münch. 2025-12-03. Transitive graphs of non-negative Ollivier-Ricci curvature have polynomial growth. https://arxiv.org/abs/2512.03968

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