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Convergence of normalized Betti numbers in nonpositive curvature

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Résumé

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 = 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)/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, 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 in [1], where we proved the same convergence result for uniformly thick sequences of compact X-manifolds.
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Dates et versions

hal-02634432 , version 1 (27-05-2020)
hal-02634432 , version 2 (02-01-2023)

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Miklos Abert, Nicolas Bergeron, Ian Biringer, Tsachik Gelander. Convergence of normalized Betti numbers in nonpositive curvature. 2021. ⟨hal-02634432v2⟩
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