The structure of arbitrary Conze-Lesigne systems

Abstract

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

69 pages, [v2]: Final version accepted for publication in Communications of the AMS