EXTENSIONS OF COCYCLES FOR HYPERFINITE ACTIONS AND APPLICATIONS

Given a countable, hyperfinite, ergodic and measure-preserving equival ence relation R on a standard probability space (X, B, mu) and an elem ent W of the normalizer N(R) of R, we investigate the problem of exten ding R-cocycles to (R) over bar-cocycles, where (R) over bar is the re lation generated by R and W. As an application, we obtain that for a B ernoulli automorphism the smallest family of natural factors in sense of [6] consists of all factors. Given an automorphism which is embedda ble in a measurable flow and a compact, metric group, we show that for a typical cocycle we cannot lift the whole flow to the centralizer of the corresponding group extension.