Extendibility of Ergodic Actions of Abelian Groups on a Measure Space

Let E be a group extension of an abelian l.c.s.c. group A by an amenable l.c.s.c. group G. We say that an ergodic action V of A is extendible to an action W of E if V (A) is isomorphic to W (A). It turns out that the extendibility property can be described in terms of cocycles over a skew product taking values in A. For topologically trivial group extensions E(G,A), we prove that the extendibility property is not generic. We give an example of R-action that is not extendible to an action of R∗+ n R. We answer the question of when two isomorphic actions of A can be extended to isomorphic actions of E(G,A). Introduction. Let A be an abelian locally compact second countable (l.c.s.c.) group and let G be an amenable l.c.s.c. group acting on A by group automorphisms. Denote by E the group extension of A by G. Then A can be identified with a normal subgroup of E. The group extension concept becomes more transparent in case of topologically trivial group extensions Ef(G,A) with f : G × G → A being a 2-cocycle. An action V of A on a measure space is called extendible to an action W of E if V (A) is isomorphic ∗Supported in part by CRDF grant 6136