The spectrum of Laplace operators and finite approximations of infinite groups

拉普拉斯算子的谱和无限群的有限近似

• 批准号: 207558863
• 负责人: Professor Dr. Roman Sauer
• 金额: --
• 依托单位: Arbeitsgruppe Topologie und Geometrische Gruppentheorie
• 依托单位国家: 德国
• 项目类别: Research Grants
• 财政年份: 2011
• 资助国家: 德国
• 起止时间: 2010-12-31 至 2014-12-31
• 项目状态: 已结题
• 来源: https://gepris.dfg.de/gepris/projekt/207558863?language=en
• 关键词: spectrum; Laplace; operators; finite; approximations

The zero-in-the-spectrum conjecture by Gromov, which is implied by the Novikov conjecture, says that the (non-reduced) l2-homology of closed aspherical manifolds vanishes. For quite a while it remained open whether this conjecture is true under much more general assumptions. One may consider two directions of generalization: a) one drops the condition of being aspherical or b) one replaces closed aspherical manifolds by classifying spaces of groups that satisfy certain finiteness conditions. The generalization a) was falsified by a counterexample of Farber and Weinberger. In this project we shall study a certain class of operad groups whose l2-homology vanishes. In particular, they provide counterexamples to b). Further, we shall study finiteness results for operad groups which unify and generalize numerous results in the literature.

由诺维科夫猜想所暗示的格罗莫夫的谱中零猜想，指出闭合非球流形的（非约化）12 -同调消失。在相当长的一段时间里，这个猜想在更普遍的假设下是否成立，仍然没有定论。我们可以考虑两种推广方向：a)放弃非球流形的条件；b)通过对满足一定有限条件的群的空间进行分类来替换闭非球流形。概括a)被Farber和Weinberger的反例证伪。在本课题中，我们将研究一类12 -同调不存在的操作群。特别是，他们为b)提供了反例。进一步，我们将研究操作群的有限性结果，这些结果统一和推广了许多文献中的结果。

项目成果

On the cohomology of weakly almost periodic group representations

关于弱几乎周期性群表示的上同调

• DOI: 10.1142/s1793525314500125
• 发表时间: 2014
• 期刊: arXiv: Group Theory
• 作者: U. Bader;C. Rosendal;R. Sauer
• 通讯作者: R. Sauer

On the growth of Betti numbers in $p$-adic analytic towers

关于 $p$-adic 分析塔中 Betti 数的增长

• DOI: 10.4171/ggd/227
• 发表时间: 2014
• 期刊: arXiv: Geometric Topology
• 作者: N. Bergeron;P. Linnell;W. Lück;R. Sauer
• 通讯作者: R. Sauer

Weak Notions of Normality and Vanishing up to Rank in L2-Cohomology

• DOI: 10.1093/imrn/rnt029
• 发表时间: 2012-06
• 期刊: International Mathematics Research Notices
• 影响因子: 1
• 作者: U. Bader;A. Furman;Roman Sauer