THE STRUCTURE OF ARBITRARY CONZE-LESIGNE SYSTEMS

Let Γ be a countable abelian group. An (abstract) Γ-system X -that is, an (abstract) probability space equipped with an (abstract) probability-preserving action of Γ -is said to be a Conze–Lesigne system if it is equal to its second Host–Kra–Ziegler factor Z²(X). The main result of this paper is a structural description of such Conze– Lesigne systems for arbitrary countable abelian Γ, namely that they are the inverse limit of translational systems G_n/Λ_n arising from locally compact nilpotent groups G_n of nilpotency class 2, quotiented by a lattice Λ_n. Results of this type were previously known when Γ was finitely generated, or the product of cyclic groups of prime order. In a companion paper, two of us will apply this structure theorem to obtain an inverse theorem for the Gowers U³(G) norm for arbitrary finite abelian groups G.