Convergence of normalized Betti numbers in nonpositive curvature

Abstract

We study the convergence of volume-normalized Betti numbers in Benjamini–Schramm convergent sequences of nonpositively curved manifolds with finite volume. In particular, we show that if X is an irreducible symmetric space of noncompact type, X≠H3, and (Mn) is any Benjamini–Schramm convergent sequence of finite-volume X-manifolds, then the normalized Betti numbers bk(Mn)∕vol(Mn) converge for all k. As a corollary, if X has higher rank and (Mn) is any sequence of distinct, finite-volume X-manifolds, then the normalized Betti numbers of Mn converge to the L2-Betti numbers of X. This extends our earlier work with Nikolov, Raimbault, and Samet, where we proved the same convergence result for uniformly thick sequences of compact X-manifolds. One of the novelties of the current work is that it applies to all quotients M=Γ∖X where Γ is arithmetic; in particular, it applies when Γ is isotropic.