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, $X\ne {\mathbb{H}}^3$, and $(M_n)$ is any Benjamini–Schramm convergent sequence of finite-volume X-manifolds, then the normalized Betti numbers $b_k(M_n)∕\mathrm{vol}(M_n)$ converge for all k.

As a corollary, if X has higher rank and $(M_n)$ is any sequence of distinct, finite-volume X-manifolds, then the normalized Betti numbers of $M_n$ converge to the $L^2$-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 $M=\mathrm{\Gamma }\setminus X$ where Γ is arithmetic; in particular, it applies when Γ is isotropic.