Groups, Actions, Metrics, Measures and Ergodic theory – GAMME

Our method lies on 6 cornerstones:

1- To allow more frequent and efficient collaborations, providing the necessary ressources;

2- Extend the perimeter of researchers working on measure group thery, L^2 invariants and geometric group theory;

3- Disseminate our knowledge internationally, through the organization of conferences, working groups and meeting in France;

4- Participation to most international conferences on these subjects, which generally take place in the USA or in Canada;

5- Hire one or two postdocs to work on these thematics (one of them finished in September 2016);

6- We finally hope that certain open problems will be solved by our new PhD students that we are planning to attract to these thematics. To this end, we plan to give advanced lectures.

During the 18 first month of our project, we have organized and sponsored various meeting and conferences around the themes of our ANR. The list is available on our webpage: math.unice.fr/~indira/GAMMEf.html.

Moreover, we have hired a postdoc, David Hume at the university of Orsay between september 2015 and september 2016. We are now getting ready to hire a new postdoc, this time at the ENS of Lyon. David Hume's recruitment was very fruitful and gave rise to 2 articles (in preparation) with Romain Tessera.

Conferences and thematic semesters are being organized and sponsored by our ANR.

The members of the ANR have published 11 articles in journals with referees during that period.

The goal of our project is to bring together prominent french, but also european mathematicians who work in areas related to measurable group theory, geometric group theory, probability and dynamics. These subjects have known dramatic development in the last fifteen years, mainly thanks to increasing integration between these different mathematical subjects. We also wish, via Post-Doc positions, conferences, and advanced courses, to disseminate the knowledge acquired worldwide, and to stimulate young mathematicians to work in this field, at the intersection of several domain of mathematics, where such a great variety of inspiring and interdisciplinary open problems are waiting to be solved.

Our scientific goals deal with various branches of geometric and measured group theory. Our project tackles five main themes of research, and will be organized over four main tasks (actually five if we take into account the organization of scientific events). Let us give a quick account of these subjects.

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. One the coordinators of this project, Romain Tessera has already obtained important results on these topics.

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 problems originate in a fundamental work of the other coordinator, Damien Gaboriau in 2002. 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 equivalence 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 generated infinite groups, some being both simple and amenable, or even Liouville. Entropy of the dynamical system might very well be related to geometric properties of the full group.

We have chosen each member of this proposal very carefully in order to find a perfect mix between a great variety of expertise and a clear connection to one or more of these themes. We hope to be successful in taking up the challenge of solving some --if not most-- of the many open problems mentioned in this proposal.