论文标题
lamplighter组引起的l^2-betti数字
L^2-Betti numbers arising from the lamplighter group
论文作者
论文摘要
我们应用了作者在先前论文中开发的建筑，以获取一个公式，该公式使我们能够计算$ \ ell^2 $ -betti数字来自一个可代表为交叉产品代数代表的组代数的家族。作为一个应用程序，我们获得了一个不合理的$ \ ell^2 $ -betti数字，该数字是由lamplighter Group algebra $ k [\ mathbb {z} _2 \ wr \ mathbb {z}] $，是$ k $ $ k $，是$ k $ $ k $ a $ k $ a $ k $ a $ k $ a $ k $ a $ k $ a的子场在复杂的同种袋中关闭。从某种意义上说，该过程具有建设性，因为人们对意识到这种非理性数字的要素有明确的描述。这扩展了Grabowski所做的工作，Grabowski首先从代数$ \ Mathbb {q} [\ Mathbb {Z} _n \ wr \ wr \ Mathbb {z}] $中计算出非理性$ \ ell^2 $ -betti编号，其中$ n \ geq 2 $是自然的数字。我们还将开发的技术应用于（广义的）里程表代数$ \ Mathcal {o}（\ overline {n}）$，其中$ \ overline {n} $是一个超自然的数字。我们计算其$*$ - 定期关闭，这使我们能够完全表征$ \ ell^2 $ -betti数字，该数字是由$ \ Mathcal {o}（\ overline {n}）$产生的。
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 $\mathcal{O}(\overline{n})$.
