The spectrum of Laplace operators and finite approximations of infinite groups

由格罗莫夫提出的零在谱猜想（英语：zero-in-the-spectrum conjecture）（英语：zero-in-the-spectrum conjecture）（英语：zero-in-the-spectrum conjecture），是由诺维科夫猜想（英语：Novikov conjecture）所隐含的，它指出闭非球面流形的（非约化）l2-同调为零。在相当长的一段时间里，这个猜想在更一般的假设下是否正确仍然是一个悬而未决的问题。我们可以考虑两个推广方向：a）一个放弃了非球面的条件，或者B）一个通过对满足某些有限性条件的群的空间进行分类来取代闭非球面流形。一般化a）被法伯和温伯格的一个反例所证伪。在这个项目中，我们将研究一类l2-同调为零的运算群。特别地，它们提供了B）的反例。此外，我们将研究有限性结果的运算群，统一和推广众多的文献中的结果。

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.