The structure of arbitrary Conze–Lesigne systems

Let Γ<!-- Γ --> Γ be a countable abelian group. An (abstract) Γ<!-- Γ --> Γ -system X X - that is, an (abstract) probability space equipped with an (abstract) probability-preserving action of Γ<!-- Γ --> Γ - is said to be a Conze–Lesigne system if it is equal to its second Host–Kra–Ziegler factor Z 2 ( X ) Z^2(X) . The main result of this paper is a structural description of such Conze–Lesigne systems for arbitrary countable abelian Γ<!-- Γ --> Γ , namely that they are the inverse limit of translational systems G n / Λ<!-- Λ --> n G_n/Λ _n arising from locally compact nilpotent groups G n G_n of nilpotency class 2 2 , quotiented by a lattice Λ<!-- Λ --> n Λ _n . Results of this type were previously known when Γ<!-- Γ --> 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 U 3 ( G ) U^3(G) norm for arbitrary finite abelian groups G G .