CONVERGENCE OF NORMALIZED BETTI NUMBERS IN NONPOSITIVE CURVATURE

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 (M_n) is any Benjamini–Schramm convergent sequence of finite-volume X-manifolds, then the normalized Betti numbers b_k(M_n)=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 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 D Γ\X where Γ is arithmetic; in particular, it applies when Γ is isotropic.