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.

A group is the mathematical notion that encapsulates the idea of symmetry, that is, of a transformation that leaves a mathematical object unchanged. Understanding the symmetries of a mathematical object is often an important step in understanding that object, making group theory a central field of modern mathematics, with ramifications in other sciences (physics, robotics, chemistry, etc.).This project was dealing with the more geometric study of abstract groups: Starting from a given group, one first tries to realize it as the symmetries of a geometric object. One then has methods from geometry available to study and understand this a priori algebraic object. This point of view, known as Geometric Group Theory, is a rather recent and active area of research, and has proved to be extremely successful in recent years, with far-reaching applications to birational geometry, algebraic topology, and 3-dimensional topology, to name but a few.A crucial problem when adopting this approach is to choose the right way to make our abstract group act as a group of symmetries of a geometric object. While several geometric tools had been developed to study so-called geometric actions (a particularly well behaved class of actions), few general tools were available to study the more general notion of cocompact action, a type of action which is ubiquitous in geometry. This projects main focus was thus to develop such tools, with applications to wide classes of groups and to the many cocompact actions naturally associated to them. As such, this project was highly successful, providing new tools applicable to vast classes of groups: The tools we developed allowed us to shed a completely new light on the Higman group, an important example in combinatorial group theory, revealing important and previously unknown informations about its structure (its automorphism group, its subgroups, etc.) Central to our approach was the understanding of a particular cocompact action on a nonpositively curved square complex. We also developed easy-to-apply criteria allowing us to determine when a given group presents a dynamical behaviour reminiscent of negative curvature, which has strong geometric implications. We were able to apply this approach to new classes of groups of various natures, such as certain groups of birational transformations, or so-called Artin groups. The new tools we developed also enabled us to understand a group through the arrangement of its various subgroups (certain substructures that are often easier to apprehend). This resulted for instance in new theorems for the large-scale classification of so-called hyperbolic groups, and to a finer understand of groups obtained trough the important method of small cancellation over a free product.