The structure of arbitrary Conze–Lesigne systems

Abstract

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