Projet ANR-06-JCJC-0099 – GGPG : Groupe de recherche de Géometrie et Probabilités dans les Groupes
Programme : Programme "Jeunes chercheuses et jeunes chercheurs" (JCJC) 2006
Année de financement : 2006
Coordinateur du projet : UNIVERSITE TOULOUSE III [PAUL SABATIER] (UNIVERSITE TOULOUSE III [PAUL SABATIER])
Partenaires : Centre national de la recherche scientifique; Centre national de la recherche scientifique; Université Claude Bernard - Lyon 1; Université de Toulouse III - Paul Sabatier
Montant : 80000.0 euros
Date de début : 2006
Durée : 36 mois
Résumé : In the last years, the use of geometric and probabilistic methods in the theory of finitely generated groups has been a great success, and solved an important number of various problems. The idea is to let a group act on a space with a good geometry, and take advantage of this geometric structure to extract properties of the group. Often the space in question is the group itself, endowed with an invariant proper metric. This approach is known as geometric group theory. In this context, asymptotic invariants, and limit actions play a crucial role. Asymptotic invariants are invariants of the large scale geometry of the considered group, or space. For example asymptotic behaviour of growth functions, asymptotic cones (Gromov-Hausdorff limits of rescalings of the considered metric space), or asymptotic dimension (a notion of dimension, due to M. Gromov, that ignores local geometry). Limit actions are induced actions on some limit space such as a boundary at infinity, or an asymptotic cone.  Applications exist and are expected in many fields, such as geometry, dynamical systems, low dimensional topology, first order logic, non standard analysis, cryptography, algorithmics, and computer science.  Let us comment briefly the most exotic applications. In 2002 Z. Sela posted a series of papers studying Tarski's problem about first order logic of free groups, and solving many question related to of actions on real trees, and variant of these, like $R^n$-trees. These objects are related to non-standard analysis.  About cryptography, let us mention that several protocols of key exchange using groups have been proposed (Ko-Lee, Anshel-Anshel-Goldfeld...) and the geometry of groups helps understanding the strenght and the weeknesses of thes protocols. In particular, a better understanding of how difficult are algorithmic problems in groups, how to efficiently choose the group, or its subsets, is relevant.   A subject of relevance for computer science is to find procedures to solve equations in diverse monoids. One can be lead to use hyperbolic geometry, in order to find efficient solutions, even for classical cases.     Among other invariants, the concept of boundary at infinity of a group is of particular interest. It is a powerful tool that has many avatars and developments. The construction of a boundary is often subtle and delicate. Practically, There are several ways to proceed. Hyperbolic geometry provides Gromov boundaries of spaces, as visual boundaries. Random walks behaviour is captured in the Poisson boundary. The Martin boundary is defined in terms of harmonic functions on the space with a probability measure. The Floyd boundary is a certain completion of some metric space. Bestvina's Z-structures are aspherical compactifications of classifying spaces for proper action... All these objects are associated to diverse structures (metric compact spaces, probability spaces, topological space endowed with a dynamical system ...) and the study of the interactions between these objects and structures has been, and is still very fruitful.    The participants of the project are all active high level researchers in fields related to these objects. They have produced significative results related to these invariants and these objects, and they have many interest in common.     Anna Erschler, Sara Brofferio, Emmanuel Breuillard, and Laurent Bartholdi, have been interested in properties of the Poisson boundaries for certain groups. They have many outstanding results in several directions, such as growth estimates, amenability, local limit theorems, etc...     Anna Erschler and Cornelia Drutu both have results on asymptotic cones of groups. A. Erschler proved, with D. Osin, that   every countable group is a subgroup of the fundamental group of the asymptotic cone of some finitely generated group. Cornelia Drutu, with M. Sapir, studied a class of asymptotic cones to prove rigidity results for relatively hyperbolic groups, an...