In this article, we will prove a full topological version of Popa’s measurable cocycle superrigidity theorem for full shifts [Popa, Cocycle and orbit equivalence superrigidity for malleable actions of $w$-rigid groups. Invent. Math. 170(2) (2007), 243–295]. Let $G$ be a finitely generated group that has one end, undistorted elements and sub-exponential divergence function. Let $H$ be a target group that is complete and admits a compatible bi-invariant metric. Then, every Hölder continuous cocycle for the full shifts of $G$ with value in $H$ is cohomologous to a group homomorphism via a Hölder continuous transfer map. Using the ideas of Behrstock, Druţu, Mosher, Mozes and Sapir [Divergence, thick groups, and short conjugators. Illinois J. Math. 58(4) (2014), 939–980; Thick metric spaces, relative hyperbolicity, and quasi-isometric rigidity. Math. Ann. 344(3) (2009), 543–595; Divergence in lattices in semisimple Lie groups and graphs of groups. Trans. Amer. Math. Soc. 362(5) (2010), 2451–2505; Tree-graded spaces and asymptotic cones of groups. Topology 44(5) (2005), 959–1058], 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, $g\geq 2,p\geq 0$; Richard Thompson groups $F,T,V$; $\text{Aut}(F_{n})$, $\text{Out}(F_{n})$, $n\geq 3$; certain (two-dimensional) Coxeter groups; and one-ended right-angled Artin groups are in our class. This partially extends the main result in Chung and Jiang [Continuous cocycle superrigidity for shifts and groups with one end. Math. Ann. 368(3–4) (2017), 1109–1132].

中文翻译：

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全班次的发散、不失真和 Hölder 连续 cocycle 超刚性

在本文中，我们将证明 Popa 的可测量 cocycle 超刚性定理的完整拓扑版本，用于全位移 [Popa, Cocycle and orbit equivalence superrigidity for malleable actions of$w$-刚性组。发明。数学。 170(2) (2007), 243–295]。让$G$是一个有限生成的群，它有一个末端、不失真的元素和次指数散度函数。让$H$是一个完整的目标群体，并承认一个兼容的双不变度量。然后，每个 Hölder 连续 cocycle 为$G$具有价值$H$通过 Hölder 连续转移映射与群同态同源。使用 Behrstock、Druţu、Mosher、Mozes 和 Sapir 的想法 [分歧、厚组和短共轭。伊利诺伊州学家数学。 58(4) (2014), 939–980; 厚度量空间、相对双曲线和准等距刚性。数学。安。 344(3) (2009), 543–595; 半单李群和群图中格的发散。反式。阿米尔。数学。社会党。 362(5) (2010), 2451–2505; 树分级空间和组的渐近锥。拓扑 44(5) (2005), 959-1058]，我们表明我们的作用群的类别很大，包括具有未失真元素的宽群和具有强厚有限阶的单端群。因此，大多数高阶连通半单李群的不可约一致格，映射类群$g$-属表面与$p$-拳头，$g\geq 2,p\geq 0$; 理查德汤普森集团$F,T,V$;$\text{Aut}(F_{n})$,$\text{输出}(F_{n})$,$n\geq 3$; 某些（二维）Coxeter 组；和单端直角阿廷组在我们班。这部分扩展了 Chung 和 Jiang 的主要结果 [Continuous cocycle superrigidity for shifts and groups with one end。数学。安。 368(3–4) (2017), 1109–1132]。