Continuous Cocycle Superrigidity for Shifts and Groups with One End

TL;DR

This paper establishes a topological cocycle superrigidity criterion for shifts over finitely generated groups with one end, linking group properties to dynamical rigidity phenomena.

Contribution

It introduces a new specification property for shifts over groups and extends cocycle superrigidity results to broader classes of systems.

Findings

Superrigidity holds iff the group has one end.

Introduces a new specification property for shifts.

Extends superrigidity results beyond $Z^d$-shifts.

Abstract

In this article, we prove that if a finitely generated group $G$ is not torsion then a necessary and sufficient condition for every full shift over $G$ has (continuous) cocycle superrigidity is that $G$ has one end. It is a topological version of the well known Popa's measurable cocycle superrigidity theorem. For the proof of sufficient condition, we introduce a new specification property for shifts over general groups which play a similar role as malleable property in the measurable setting. This new specification property is good enough for us to extend the method using homoclinic equivalence relation that was introduced by Klaus Schmidt to study cocycle rigidity for $\mathbb{Z}^d$-shifts. Indeed, in this direction we prove this superrigidity result for more certain general systems. And for the converse, we apply Specker's characterization for ends of groups via the associated first cohomology groups to get the result. Finally, combining our results with a result of Xin Li, we have an application in continuous orbit equivalence rigidity.