Cocompact actions on polyhedral complexes

Geometric group theory is the study of groups through their actions on spaces endowed with a suitable geometry. A central problem of geometric group theory which is at the heart of this project is the following: Given a group action on a simply-connected polyhedral complex, is it possible to deduce a property for the group -- algebraic, geometric, analytic or algorithmic -- out of the same property for the stabilisers of faces, provided one imposes conditions on the geometry of the complex and the way it is acted upon? While such so-called combination problems have been extensively studied in the context of groups acting on trees, the situation has remained mostly unaddressed as of now in the case of non- proper actions on complexes of dimension greater than 1. The major goal of this project is to develop tools to tackle such combination problems in full generality from the point of view of complexes of groups, with a view towards two properties at the forefront of geometric group theory: the hyperbolicity of a group and the existence of a non-positively curved cubulation. We plan to obtain very general combination theorems for hyperbolic groups, which will be used to construct new examples of hyperbolic groups with unusual and surprising properties. The point of view adopted in this project will also pave the way for an in-depth understanding of the geometry and the residual properties of groups which are obtained through various forms of high- dimensional small cancellation ranging from small cancellation over a complex of groups to iterated small cancellation. This will lead, in particular, to a better understanding of the yet elusive Rips-Segev groups, with applications to the celebrated Kaplansky`s zero-divisor conjecture. This project is the first to bring cutting-edge tools from the theory of high-dimensional complexes of groups to study groups through their non-proper actions in full generality. It will undoubtly have a large impact in the field as such actions are ubiquitous in geometric group theory.

No additional funding sources recorded.