Profinite perspectives on L2-cohomology

从一个剩余有限群的有限子群中可以得到多少关于它的信息？这个问题已经在不同的背景下提出，但它是特别有趣的简单李群格。 在这个研究项目中，我们想研究在何种程度上，群，特别是格的L2-上同调性质是由它们的profinite完备化决定的。后者也将作为动力系统进行研究时，赋予翻译行动。作为几何结果，我们将确定欧拉特征线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确定的情况。我们将研究L2-上同调的代数逼近性质，这将在代数特征值性质方面提供新的见解。目标之一是建立左序群的Atiyah猜想，我们将利用格的刚性性质来研究L2-上同调的profinite方面。在这个项目的过程中，我们计划进一步研究在多大程度上的结果格扩展到更大的类的剩余有限群。例如，虽然L2-无环性是高阶格的profinite不变量，但这是否对任意剩余有限群成立是未知的。 这条线的调查联系我们的项目，以组理论的问题，出现在试图构建反例。

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.