2019/09/12 by Luca F. Di Cerbo, Di Cerbo, Luca F., Mark Stern +1
Mathematics · #Analytic and geometric function theory #Differential Geometry (math.DG) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Geometric Topology (math.GT) #Nonlinear Partial Differential Equations
paper · pdf · doi:10.48550/arxiv.1909.05634
openalex publication_date 2019/09/12 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We study Betti numbers of sequences of Riemannian manifolds which Benjamini-Schramm converge to their universal covers. Using the Price inequalities we developed elsewhere, we derive two distinct convergence results. First, under a negative Ricci curvature assumption and no assumption on sign of the sectional curvature, we have a convergence result for weakly uniform discrete sequences of closed Riemannian manifolds. In the negative sectional curvature case, we are able to remove the weakly uniform discreteness assumption. This is achieved by combining a refined Thick-Thin decomposition together with a Moser iteration argument for harmonic forms on manifolds with boundary.