Geometric invariants of discrete and locally compact groups

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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.