Geometric invariants of discrete and locally compact groups

群是描述对称性的代数对象。许多重要的群（甚至是有限对象的对称性）都是无限的。这就提出了一个问题，一个给定的群体是否可以用有限的信息量来描述。这个问题在几个程度上出现，并导致了一系列所谓的有限性属性。西格玛不变量细化了有限性属性所捕获的信息：它们在几何意义上是有方向的。我们的目标是更好地理解一般的Sigma不变量，并完全确定某些有趣的和重要的群的Sigma-不变量。该项目的一个目标是将理论扩展到拓扑群。这与算术群（数论术语中定义的群）有关，这是该项目的另一个兴趣中心。最后，我们想进一步研究阿廷群的Σ-不变量，因为它们在过去被证明是特别有趣的。

Groups are algebraic objects that describe symmetries. Many important groups (even symmetries of finite objects) are infinite. This raises the question whether a given group can be described by a finite amount of information. The question comes in several degrees and leads to a sequence of so-called finiteness properties.Sigma-invariants refine the information captured by finiteness properties: they are in a geometric sense directional. Our goal is to better understand Sigma invariants in general and to completely determine the Sigma-invariants of certain interesting and important groups.So far the theory of Sigma-invariants dealt exclusively with discrete groups. One goal of the project is to extend the theory to topological groups. This is related to arithmetic groups (groups defined in number-theoretic terms), another center of interest of the project. Finally we want to further investigate the Sigma-invariants of Artin groups since they have proven particularly interesting in the past.