Divergence, Undistortion and H\"older Continuous Cocycle Superrigidity for Full Shifts

Abstract

In this article, we will prove a full topological version of Popa's measurable cocycle superrigidity theorem for full shifts. More precisely, we prove that every H\"older continuous cocycle for the full shifts of every finitely generated group $G$ that has one end, undistorted elements and sub-exponential divergence function is cohomologous to a group homomorphism via a H\"older continuous transfer map if the target group is complete and admits a compatible bi-invariant metric. Using the ideas of Behrstock, Dru\c {t}u, Mosher, Mozes and Sapir, we show that the class of our acting groups is large including wide groups having undistorted elements and one-ended groups with strong thick of finite orders. As a consequence, irreducible uniform lattices of most of higher rank connected semisimple Lie groups, mapping class groups of $g$-genus surfaces with $p$-punches, $g\geq 2, p\geq 0$, Thompson group, $Aut(F_n)$, $Out(F_n)$, $n\geq3$, certain (2 dimensional)-Coxeter groups, and one-ended right-angled Artin groups are in our class. This partially extends the main result in our previous paper.