Justin Salez: Sparse expanders have negative curvature

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 Naor and Milman and publicized by Ollivier (2010).
To establish this, we work 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 spectral expansion are incompatible “at infinity”. We then transfer this result to finite graphs via local weak convergence. The same approach applies to the Bakry-Émery curvature condition CD(0, ∞), thereby settling a recent conjecture of Cushing, Liu and Peyerimhoff (2019).

After the talk there will be an open problem session for discussion, highly recommended for researchers on the DYNASNET project.