Continuous cocycle superrigidity for shifts and groups with one end

Abstract

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 (Cocycle and orbit equivalence superrigidity for malleable actions of w-rigid groups. Invent. Math. 170(2):243–295, 2007). For the proof of sufficiency, we introduce a new specification property for shifts over general groups which plays a similar role as malleable property in the measurable setting. This new specification property is good enough for us to extend the method of using homoclinic equivalence relation that was introduced by Schmidt (Pac J Math 170(1):237–269, 1995) to study cocycle rigidity for $${{\mathbb {Z}}}^d$$ -shifts. Indeed, in this direction we prove this superrigidity result for certain more general systems. And for the necessary condition, we apply Specker’s characterization for ends of groups via the associated first cohomology groups to get the result. Finally, combining our results with Li (Ergod Theory Dyn Syst. arXiv:1503.01704 , Theorem 1.6), we have an application in continuous orbit equivalence rigidity.