Our project tackles five main themes of research, in various branches of geometric and measured group theory:
The first theme deals with the interaction between the structure of locally compact groups and its geometric and ergodic properties. It is important to enhance that our main goal here is either to explore specific features of non-discrete groups or to exploit results on locally compact groups to obtain new insights on finitely generated ones.
Our second theme is related to measured equivalence relations and mainly focuses on two long standing questions: the fixed Price problem and the cost vs first L2 Betti number. These two problems have strong consequences in Measure Equivalence theory, L2 Invariants theory, von Neumann Algebras and in Percolation on graphs.
Instead of a theme, our third part gathers various probabilistic methods in ergodic theory: percolation, invariant random subgroups, and Poisson boundary. These topics form a broad subject which is intimately related to the previous theme, however also coming with their own fascinating open problems. We expect them to shed light on ergodic theory, and vice versa.
The fourth theme very interestingly links together many of the previous themes as it combines geometry and measured theory in a very intricate way. Integrable measure equiv- alence strengthens measure equivalence by taking into account the large-scale geometry of the group via some first moment condition imposed on the coupling. Many important and surprising rigidity results for amenable groups, and for lattices in simple Lie groups of rank 1 have recently brought this subject to the forefront of research in geometric group theory.
Finally, our last theme has a more topological dynamics flavor. The idea of soficity takes its origins in the work of Gromov, who aimed to formulate a finite approximation property for groups that encompasses both amenability and residual finiteness. A recent breakthrough in topological dynamics is a definition of entropy invented by Bowen for actions of sofic groups. The notion of topological full group of a topological dynamical system has become prominent recently as it provides a whole new family of finitely gener- ated infinite groups, some being both simple and amenable, or even Liouville. The entropy of the dynamical system might be related to geometric properties of the full group.