Rigidity, weak mixing, and recurrence in abelian groups

TL;DR

This paper extends the understanding of rigidity, weak mixing, and recurrence from integer actions to general abelian groups, providing new constructions and insights into these phenomena in more complex group settings.

Contribution

It generalizes key results about rigidity sequences from $ ext{Z}$-actions to all countable abelian groups, introducing new techniques for groups with multiple generators.

Findings

Rigidity sequences for $ ext{Z}$-actions extend to weakly mixing systems.

Existence of sequences that are both rigidity sequences and recurrence sets in abelian groups.

New constructions of rigidity sequences in torsion groups.

Abstract

The focus of this paper is the phenomenon of rigidity for measure-preserving actions of countable discrete abelian groups and its interactions with weak mixing and recurrence. We prove that results about $\mathbb{Z}$-actions extend to this setting: 1. If $(a_n)$ is a rigidity sequence for an ergodic measure-preserving system, then it is a rigidity sequence for some weakly mixing system. 2. There exists a sequence $(r_n)$ such that every translate is both a rigidity sequence and a set of recurrence. The first of these results was shown for $\mathbb{Z}$-actions by Adams, Fayad and Thouvenot, and Badea and Grivaux. The latter was established in $\mathbb{Z}$ by Griesmer. While techniques for handling $\mathbb{Z}$-actions play a key role in our proofs, additional ideas must be introduced for dealing with groups with multiple generators. As an application of our results, we give several new constructions of rigidity sequences in torsion groups. Some of these are parallel to examples of rigidity sequences in $\mathbb{Z}$, while others exhibit new phenomena.