15 March 2023 Convergence of normalized Betti numbers in nonpositive curvature
Miklos Abert, Nicolas Bergeron, Ian Biringer, Tsachik Gelander
Author Affiliations +
Duke Math. J. 172(4): 633-700 (15 March 2023). DOI: 10.1215/00127094-2022-0029


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, XH3, 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=ΓX where Γ is arithmetic; in particular, it applies when Γ is isotropic.


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Miklos Abert. Nicolas Bergeron. Ian Biringer. Tsachik Gelander. "Convergence of normalized Betti numbers in nonpositive curvature." Duke Math. J. 172 (4) 633 - 700, 15 March 2023. https://doi.org/10.1215/00127094-2022-0029


Received: 22 February 2022; Revised: 24 February 2022; Published: 15 March 2023
First available in Project Euclid: 8 February 2023

MathSciNet: MR4557758
zbMATH: 07684349
Digital Object Identifier: 10.1215/00127094-2022-0029

Primary: 53C23
Secondary: 57T15

Keywords: Benjamini Schramm convergence , Betti number , locally symmetric space

Rights: Copyright © 2023 Duke University Press


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Vol.172 • No. 4 • 15 March 2023
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