The structure of arbitrary Conze-Lesigne systems

TL;DR

This paper characterizes the structure of Conze-Lesigne systems for any countable abelian group, showing they are inverse limits of systems from nilpotent groups of class 2, extending previous results.

Contribution

It provides a general structural description of Conze-Lesigne systems for arbitrary countable abelian groups, generalizing earlier known cases.

Findings

Conze-Lesigne systems are inverse limits of translational systems from nilpotent groups of class 2.

The structure theorem applies to all countable abelian groups, not just finitely generated or cyclic groups.

This result facilitates further applications, such as inverse theorems for Gowers norms.

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 $G$.