Groupes sofiques: algèbre, analyse et dynamique
| Responsable du projet |
Alain Valette
Nicolas Monod Goulnara Arjantseva |
| Collaborateur |
Adam Timar
Sabine Burgdorf Liviu Paunescu Christophe Gippa Swiatoslaw Gal Valerio Capraro |
| Résumé |
Sofic groups, by their nature, break frontiers between various areas
of pure mathematics (geometric group theory, dynamical systems,
operator algebras). This is a very promising class of groups whose
recent genesis and modern understanding of geometric, asymptotic,
and algebraic structures can generate new examples and theories in
analytic properties of groups. It appears to be necessary to
develop their study further and investigate connections with
geometric group theory, topology, and dynamics. Analysis on
discrete infinite groups is establishing itself as a new branch of
group theory, with techniques borrowing from metric geometry,
operator algebras, and harmonic analysis. Since 1980, Gromov is
treating finitely generated groups as metric spaces, and a wealth
of results by Gromov and followers show that a large part of the
algebraic structure is captured by metric properties. In
particular, Gromov introduced several asymptotic invariants of
groups, i.e. quasi-isometry invariants, robust with respect to
"local" perturbations and depending only on the
"large-scale" geometry of the group. Many of these
invariants are analytic in nature. The main goal of this project is
to work out the precise relations between soficity and several
group-theoretical properties which emerged recently, some
analytic/algebraic (Haagerup property, C*-exactness,
hyperlinearity,...), some metric (word hyperbolicity, Yu's property
(A),...). Some of these properties imply positive results towards
deep conjectures on group algebras, like the Baum-Connes
conjecture, and our second objective is to contribute to some of
these conjectures, by providing new classes of groups satisfying
them. In this project we will focus on groups having geometric
content, as a natural setup for doing analysis: hyperbolic groups,
groups acting on trees, groups acting on Hilbert spaces, discrete
subgroups of Lie groups. |
| Mots-clés |
Sofic groups, Hyperlinear groups, Surjunctive groups, Haagerup property (a-T-menability), C*-exactness, Uniform embeddings into Hilbert spaces |
| Type de projet | Recherche fondamentale |
| Domaine de recherche | Mathématiques |
| Source de financement | FNS - Sinergia |
| Etat | Terminé |
| Début de projet | 1-5-2010 |
| Fin du projet | 30-4-2013 |
| Budget alloué | 841'014.00 |
| Autre information |
http://p3.snf.ch/projects-130435# |
| Contact | Alain Valette |