Geometric Group Theory for Locally Compact Groups
Geometric group theory studies groups using geometric, topological, and metric tools. The basic idea is to regard a group not only as an algebraic object but also as geometric one, through its actions on spaces—such as Cayley graphs. The aim is to analyze the relationship between the algebraic structure of a group and its geometry, in order to address questions about algorithmic problems (such as the word problem), cohomology, and more. A central theme is the study of large-scale properties—those invariant under quasi-isometries—such as growth, accessibility, and finiteness conditions. Among the most studied groups in this area are hyperbolic groups, automatic groups, and groups acting on trees.In recent decades, this theory has been extended to locally compact groups (that is, Hausdorff topological groups with compact neighborhoods of the identity). In this context, many tools from the discrete theory do not apply directly, and more sophisticated techniques are required. This extension is important because it allows the study of many non-discrete groups (such as p-adic groups and isometry groups), and makes it possible to explore new phenomena that do not arise in the theory of abstract groups. We are mostly interested in the study of homological finiteness properties of totally disconnected locally compact groups, which is an important emerging research area.