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Abstract

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 Z 2 (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 𝑈 3 (𝐺) norm for arbitrary finite abelian groups 𝐺.