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Sparse expanders have negative curvature

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Abstract

We prove that bounded-degree expanders with non-negative Ollivier–Ricci curvature do not exist, thereby solving a long-standing open problem suggested by A. Naor and E. Milman and publicized by Y. Ollivier (2010). In fact, this remains true even if we allow for a vanishing proportion of large degrees, large eigenvalues, and negatively-curved edges. Moreover, the same conclusion applies to the Bakry–Émery curvature condition CD\((0,\infty )\), thereby settling a recent conjecture of D. Cushing, S. Liu and N. Peyerimhoff (2019). To establish those results, we work directly at the level of Benjamini–Schramm limits, and exploit the entropic characterization of the Liouville property on stationary random graphs to show that non-negative curvature and uniform expansion are incompatible “at infinity”. We then transfer this conclusion to finite graphs via local weak convergence. Our approach also shows that the class of finite graphs with non-negative curvature and degrees at most d is hyperfinite, for any fixed \(d\in {\mathbb {N}}\).

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Acknowledgements

The author warmly thanks M. Abért, I. Benjamini, N. Curien, D. Cushing, S. Liu, R. Lyons, F. Münch, P. Pansu, G. Pete, and two anonymous referees for many wonderful comments and references. This work was partially supported by Institut Universitaire de France.

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  1. CEREMADE, CNRS, UMR 7534, Université Paris-Dauphine, PSL University, 75016, Paris, France

    Justin Salez

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Salez, J. Sparse expanders have negative curvature. Geom. Funct. Anal. 32, 1486–1513 (2022). https://doi.org/10.1007/s00039-022-00618-3

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