3 - Groups of Automorphisms of Measure Space and Weak Equivalence of Cocycles

Summary

This survey is devoted to a brief exposition of results proved mainly in papers [BG4, BG5, GS3, GS4]. We study ergodic countable approximately finite groups Γ of non-singular automorphisms of a measure space (X, μ) and cocycles α : X x Γ → G taking values in a l.c.s.c group G. The concept of weak equivalence of pairs (Γ, α) (which can be treated as a generalization of orbit equivalence of countable automorphism groups) was introduced and studied in these articles. All pairs (Γ, α) and (Γ, α × ρ) can be classified by associated Mackey actions of G and G × R where ρ is the Radon-Nikodym cocycle of Γ. The structure of cocycles up to weak equivalence is described. It is shown that the proved results can be applied to the solution of the outer conjugacy problem. Other applications of weak equivalence of cocycles are also considered.

Acknowledgment. The author was supported in part by INTAS-97 grant.

Introduction

Cocycles have been the subject of extensive investigations in the ergodic theory during the last thirty years. They appear naturally under solution of many problems because a cocycle over a single transformation of a measure space is represented by a measurable function taking values in a group G. An important role of cocycles is provided by the possibility to construct the new group actions which reflect the basic properties of the initial dynamical system.