Convergence of normalized Betti numbers in nonpositive curvature

Abért, Miklós and Bergeron, Nicolas and Biringer, Ian and Gelander, Tsachik (2023) Convergence of normalized Betti numbers in nonpositive curvature. DUKE MATHEMATICAL JOURNAL. ISSN 0012-7094

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Abstract

We study the convergence of volume-normalized Betti numbers in Benjamini-Schramm convergent sequences of non-positively curved manifolds with finite volume. In particular, we show that if X is an irreducible symmetric space of noncompact type, X 6 = 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, the normalized Betti numbers of Mn converge to the L2 Betti numbers of X. This extends our earlier work with Nikolov, Raimbault and Samet in [1], 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.

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