Approximating L2‐invariants and the Atiyah conjecture
Approximating L2‐invariants and the Atiyah conjecture
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近似 L2 不变量和 Atiyah 猜想
DOI：
10.1002/cpa.10076
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发表时间：
2001
影响因子：
3
通讯作者：
Stuart Yates
中科院分区：
文献类型：
--
作者：
J. Dodziuk;P. Linnell;V. Mathai;T. Schick;Stuart Yates
令 G 为无挠离散群，令 ℚ 表示 ℂ 中的代数数域。我们证明，如果 G 位于某类群 D 中，则 ℚG 满足 Atiyah 猜想，该群特别包含所有剩余无挠初等或剩余自由的群。这个结果意味着 ℂG 中不存在非平凡的零因子。该声明依赖于 ℚG 上 L2-Betti 数的新近似结果，这是本文所做工作的核心。本文中的另一组结果涉及有限 CW 复形的任何法向覆盖空间上的 L2-cochains 上的组合拉普拉斯特征值的某些数论性质。每当覆盖变换群服从或在 Linnell 类中时，我们就确定不存在作为超越数的特征值。当覆盖变换群是残差有限或更一般地在某个大的自举类中时，我们还确定不存在作为刘维尔超越数的特征值。 © 2003 Wiley 期刊公司。
Let G be a torsion‐free discrete group, and let ℚ denote the field of algebraic numbers in ℂ. We prove that ℚG fulfills the Atiyah conjecture if G lies in a certain class of groups D, which contains in particular all groups that are residually torsion‐free elementary amenable or are residually free. This result implies that there are no nontrivial zero divisors in ℂG. The statement relies on new approximation results for L2‐Betti numbers over ℚG, which are the core of the work done in this paper. Another set of results in the paper is concerned with certain number‐theoretic properties of eigenvalues for the combinatorial Laplacian on L2‐cochains on any normal covering space of a finite CW complex. We establish the absence of eigenvalues that are transcendental numbers whenever the covering transformation group is either amenable or in the Linnell class ?. We also establish the absence of eigenvalues that are Liouville transcendental numbers whenever the covering transformation group is either residually finite or more generally in a certain large bootstrap class ?. © 2003 Wiley Periodicals, Inc.