Convergence of normalized Betti numbers in nonpositive curvature

Miklos Abert (1) , Nicolas Bergeron (2) , Ian Biringer (3) , Tsachik Gelander (4)

1 CEU - Central European University [Budapest, Hongrie]
2 DMA - Département de Mathématiques et Applications - ENS Paris
3 BC - Boston College
4 Institute of Mathematics

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.

Domaines: Topologie géométrique [math.GT]; Mathématiques [math]

Miklos Abert, Nicolas Bergeron, Ian Biringer, Tsachik Gelander. Convergence of normalized Betti numbers in nonpositive curvature. 2021. ⟨hal-02634432v1⟩