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ANR funded project

Défi de tous les savoirs (DS10) 2014
Projet GAMME

Groups, Actions, Metrics, Measures and Ergodic theory

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.

Partners

UMPA Unité de Mathématiques Pures et Appliquées

UPSud / LMO UNIVERSITE PARIS-SUD / Laboratoire de Mathématique d'Orsay

ANR grant: 396 776 euros
Beginning and duration: octobre 2014 - 48 mois

 

ANR Programme: Défi de tous les savoirs (DS10) 2014

Project ID: ANR-14-CE25-0004

Project coordinator:
Monsieur Romain Tessera (UNIVERSITE PARIS-SUD / Laboratoire de Mathématique d'Orsay)

 

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The project coordinator is the author of this abstract and is therefore responsible for the content of the summary. The ANR disclaims all responsibility in connection with its content.