Analysis and Probability Seminar

​Eusebio Gardella, MV, Chalmers/GU: The classification problem for free ergodic actions


One 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. No previous knowledge on group amenability will be assumed, and all relevant definitions will be introduced in the course of the presentation.

The seminar will be held on campus, at MV:L14, and via zoom. Members of the department will receive the link and password via mail. Others interested are welcome to attend and can get the link by contacting one of the organisers (please inform us who you are).

​Organisers: Jakob Björnberg (jakobbj@chalmers.se), Erik Broman (broman@chalmers.se) and Genkai Zhang (genkai@chalmers.se). Please contact us if you wish to be informed of future seminars.
Kategori Seminarium
Plats: MV:L14, Chalmers tvärgata 3, and online
Tid: 2021-10-05 13:15
Sluttid: 2021-10-05 14:15

Sidansvarig Publicerad: fr 01 okt 2021.