Convergence of normalized Betti numbers in nonpositive curvature

2018/11/06 by Abert, Miklos, Bergeron, Nicolas, Biringer, Ian +1
#FOS: Mathematics #Geometric Topology (math.GT)

paper · doi:10.48550/arxiv.1811.02520

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 ≠ \mathbb 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, 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.

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