由灯夫群产生的L^2-Betti数
L^2-Betti numbers arising from the lamplighter group
AI总结：
本文利用交叉积代数构造公式，计算灯夫群代数及广义里程计代数的无理ℓ²-Betti数，扩展了Grabowski的结果，并完全刻画了里程计代数的ℓ²-Betti数集合。
摘要：
我们应用作者先前论文中发展的一种构造，以获得一个公式，该公式使我们能够计算来自一族可表示为交叉积代数的群代数的ℓ²-Betti数。作为应用，我们获得了来自灯夫群代数K[Z₂≀Z]的一整族无理ℓ²-Betti数，其中K是复数域的一个在复共轭下封闭的子域。这个过程是构造性的，即对于实现这些无理数的元素有显式描述。这扩展了Grabowski的工作，他首次从代数Q[Zₙ≀Z]（其中n≥2为自然数）计算了无理ℓ²-Betti数。我们还将所发展的技术应用于（广义）里程计代数O(n̄)，其中n̄是一个超自然数。我们计算了它的*-正则闭包，这使我们能够完全刻画由O(n̄)产生的ℓ²-Betti数的集合。
英文摘要：
We apply a construction developed in a previous paper by the authors in order to obtain a formula which enables us to compute $\ell^2$-Betti numbers coming from a family of group algebras representable as crossed product algebras. As an application, we obtain a whole family of irrational $\ell^2$-Betti numbers arising from the lamplighter group algebra $K[\mathbb{Z}_2 \wr \mathbb{Z}]$, being $K$ a subfield of the complex numbers closed under complex conjugation. This procedure is constructive, in the sense that one has an explicit description of the elements realizing such irrational numbers. This extends the work made by Grabowski, who first computed irrational $\ell^2$-Betti numbers from the algebras $\mathbb{Q}[\mathbb{Z}_n \wr \mathbb{Z}]$, where $n \geq 2$ is a natural number. We also apply the techniques developed to the (generalized) odometer algebra $\mathcal{O}(\overline{n})$, where $\overline{n}$ is a supernatural number. We compute its $*$-regular closure, and this allows us to fully characterize the set of $\ell^2$-Betti numbers arising from $\mathcal{O}(\overline{n})$.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
