Duke Math. J. 172 (4), 633-700, (15 March 2023) DOI: 10.1215/00127094-2022-0029
Miklos Abert, Nicolas Bergeron, Ian Biringer, Tsachik Gelander
KEYWORDS: Benjamini Schramm convergence, Betti number, locally symmetric space, 53C23, 57T15
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, , and is any Benjamini–Schramm convergent sequence of finite-volume X-manifolds, then the normalized Betti numbers converge for all k.
As a corollary, if X has higher rank and is any sequence of distinct, finite-volume X-manifolds, then the normalized Betti numbers of converge to the -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 where Γ is arithmetic; in particular, it applies when Γ is isotropic.