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

TL;DR

This paper proves a topological version of cocycle superrigidity for full shifts of certain groups, showing that H"older continuous cocycles are cohomologous to homomorphisms under specific conditions.

Contribution

It extends Popa's measurable cocycle superrigidity to a topological setting for a broad class of groups with specific geometric properties.

Findings

H"older continuous cocycles are cohomologous to homomorphisms for the considered groups.

The class of groups includes many important groups such as lattices, mapping class groups, and right-angled Artin groups.

The results generalize previous work by incorporating topological aspects and broader group classes.

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.