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## Convergence of normalized Betti numbers in nonpositive curvature. (arXiv:1811.02520v2 [math.GT] UPDATED)

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 \neq \mathbb 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, where we proved the same convergence result for uniformly thick sequences of compact \$X\$-manifolds.查看全文

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 你的名字 留空匿名提交 你的Email或网站 用户可以联系你 标题 简单描述 内容 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 \neq \mathbb 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, where we proved the same convergence result for uniformly thick sequences of compact \$X\$-manifolds.
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