关于正规生成与第一$\ell^2$-Betti数的注记
A note on normal generation and the first $\ell^2$-betti number
AI总结：
本文构造了满足特定$\ell^2$-Betti数与正规秩关系的可数无挠群，证伪了Osin和Thom关于无挠离散群第一$\ell^2$-Betti数上界的猜想。
AI中文摘要：
2011年，Osin和Thom猜想，无挠离散群的第一$\ell^2$-Betti数上界为该群的正规秩减1。该猜想对群论与拓扑学的若干基础问题有出人意料的推论，包括关于完全群的Wiegold问题、Levin猜想、Kervaire猜想的无挠情形，以及Whitehead非球面性猜想的一个重要特例。本文中，我们对每个自然数$n$构造了一个可数无挠群$\Gamma_n$，满足$\beta^{(2)}_1(\Gamma_n)=n$，且其正规秩$n(\Gamma_n)$等于1，这一结果证伪了上述猜想。我们构造的反例是局部自由的，因此也是局部可定向的，但它们并非有限生成。
英文摘要：
In $2011$, Osin and Thom conjectured that the first $\ell^2$-Betti number of a torsion-free discrete group is bounded above by the normal rank of the group minus one. The conjecture has surprising consequences for some fundamental problems in group theory and topology. These include the Wiegold problem on perfect groups, the Levin conjecture, the torsion-free case of the Kervaire conjecture, and an important special case of the Whitehead asphericity conjecture. In this article, we construct for each $n\in \mathbb{N}$ a countable torsion-free group $Γ_n$ such that $β^{(2)}_1(Γ_n)=n$ and so that the normal rank, $n(Γ_n)$, equals one. This disproves the conjecture. Our counterexamples are locally free and hence locally indicable. However, they are not finitely generated.
