Profinite perspectives on L2-cohomology

一个残差有限群可以从它的有限商中恢复多少信息？这个问题已经在不同的背景下被提出，但对于简单李群中的格来说，它特别有趣。在这个研究项目中，我们想要研究群，特别是格的l2 -上同调性质是由它们的无限补全决定的。当赋予翻译作用时，后者也将作为动力系统进行研究。作为几何结果，我们将确定欧拉特征的符号或局部对称空间的体积由其有限片覆盖的甲板变换群决定的情况。我们将研究l2 -上同调的代数近似性质，这将给代数特征值性质提供新的见解。一个目标是建立左可序群的Atiyah猜想。我们将开始利用格的刚性性质研究l2 -上同调的无限方面。在这个项目的过程中，我们进一步计划研究格的结果在多大程度上扩展到更大的剩余有限群类。例如，虽然l2 -不环性是高阶格的定不变量，但对于任意剩余有限群是否成立是未知的。这条调查路线将我们的项目与试图构建反例时出现的群理论问题联系起来。

How much information on a residually finite group can one recover from its finite quotients? This question has been raised in diverse contexts but it is particularly intriguing for lattices in simple Lie groups. In this research project, we want to investigate to what extend L2-cohomological properties of groups, in particular lattices, are determined by their profinite completions. The latter will also be investigated as dynamical systems when endowed with translation actions. As geometric outcomes, we will identify situations in which the sign of the Euler characteristic or the volume of a locally symmetric space are determined by the deck transformation groups of its finite sheeted coverings. We will investigate algebraic approximation properties of L2-cohomology, which should give new insights in the algebraic eigenvalue property. One goal is to establish the Atiyah conjecture for left-orderable groups.We will start to investigate profinite aspects of L2-cohomology using rigidity properties of lattices. In the course of this project we further plan to investigate in how far results for lattices extend to larger classes of residually finite groups. For instance, while L2-acyclicity is a profinite invariant of higher rank lattices, it is unknown whether this holds true for arbitrary residually finite groups. This line of investigations links our project to group theoretic problems which arise in the attempt to construct counterexamples.