Séminaires : Séminaire d'Algèbres d'Opérateurs

Equipe(s) : ao,
Responsables :Pierre Fima, François Le Maître, Romain Tessera
Email des responsables :
Salle : 2015
Adresse :Sophie Germain

Orateur(s) Eusebio Gardella - WWU Münster,
Titre The classification problem for free ergodic actions
Horaire14:00 à 15:00
Diffusion https://bigbluebutton3.imj-prg.fr/b/fra-j6k-9fw
RésumeOne of the basic problems in Ergodic Theory is to determine when two measure-preserving actions of a group on the atomless Borel probability space are orbit equivalent. When the group is amenable, classical results of Dye and Ornstein-Weiss show that any two such actions are orbit equivalent. Thus, the question is relevant only in the non-amenable case. In joint work with Martino Lupini, we showed that for every nonamenable countable discrete group, the relations of conjugacy and orbit equivalence of free ergodic actions are not Borel, thereby answering questions of Kechris. This means that there is in general no method, or uniform procedure, that allows us to determine when two actions of a nonamenable group are conjugate/orbit equivalent. It is a non-classification result, which rules out the existence of any classification theorems which use "nice" (Borel) invariants. The statement about conjugacy also solves the nonamenable case of Halmos' conjugacy problem in Ergodic Theory, originally posed in 1956 for ergodic transformations. The main conceptual innovation is the notion of property (T) for triples of groups, for which a cocycle superrigidity theorem à la Popa can be established. In combination with induction methods developed by Epstein, this is used to obtain a large family of free ergodic actions of the given nonamenable group which have pairwise distinct 1-cohomology groups.
AdresseSophie Germain