On Cocycle Superrigidity Problem for Algebraic Actions
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In this thesis, we study cocycle superrigidity problem for algebraic actions in both measurable setting and continuous setting. After introduction to this topic in the first chapter, we review some important concepts used in this thesis in the second chapter. In Chapter 3, we study the (measurable) T -valued cohomology groups of algebraic actions. We prove that for a weakly mixing algebraic action σ: G → ( X , ν) the n -cohomology group H n ( G → X , T ), after quotienting out the natural subgroup H n ( G, T ), contains H n ( G, Xˆ ) as a natural subgroup for n = 1. If we further assume the diagonal actions σ 2 , σ 4 are T -cocycle superrigid and H 2 ( G, Xˆ ) is torsion free as an abelian group, then the above also holds true for n = 2. Applying it for principal algebraic actions when n = 1, we show that H 2 ( G, Z G ) is torsion free as an abelian group when G has property (T) as a direct corollary of Sorin Popa's cocycle superrigidity theorem; we also use it (when n = 2) to answer, negatively, a question of Sorin Popa on the 2nd cohomology group of Bernoulli shift actions of property (T) groups. This chapter is based on my paper . In Chapter 4, we study the continuous cocycle superrigidity problem for shifts. For example, we show that for a finitely generated non-torsion group G, the full shift ( G, A G ) is an H -continuous cocycle superrigid action for any finite set A and any discrete group H if and only if G has one end. Finally, combining our results with a result of Xin Li, we have an application in continuous orbit equivalence rigidity theory. This chapter is based on the paper , which is joint work with Nhan-Phu Chung.