The spectrum of Laplace operators and finite approximations of infinite groups

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。

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.