论文标题
从高级晶格到$ \ mathrm {out}（f_n）$
Cocycle superrigidity from higher rank lattices to $\mathrm{Out}(F_N)$
论文作者
论文摘要
我们证明了从高级晶格到$ \ mathrm {out}（f_n）$的cocycles的刚度结果，更笼统地是无扭转双曲线组的外部自动形态组。更确切地说，让$ g $是在本地字段上连接的较高等级简单的代数组的产品，或者是这种产品中的晶格。令$ g \ curvearrowright x $为标准概率空间上的千古量度保护动作，让$ h $为无扭转的双曲线组。我们证明，每个Borel Cocycle $ g \ times x \ to \ mathrm {out}（h）$与cocycle共同出现，并在有限的子组中，$ \ mathrm {out}（out}（h）$。这提供了Farb的定理 - Kaimanovich（Masur and Bridson）的动态版本，并断言，从$ G $到有限型表面的映射类组或自由组的外部自动形态组的映射类组都有有限的图像。 主要的新几何工具是一个barycenter图，它将（相对）自由因子图的边界中的每个点相关联，一个有限的（相对）自由分组。
We prove a rigidity result for cocycles from higher rank lattices to $\mathrm{Out}(F_N)$ and more generally to the outer automorphism group of a torsion-free hyperbolic group. More precisely, let $G$ be either a product of connected higher rank simple algebraic groups over local fields, or a lattice in such a product. Let $G\curvearrowright X$ be an ergodic measure-preserving action on a standard probability space, and let $H$ be a torsion-free hyperbolic group. We prove that every Borel cocycle $G\times X\to\mathrm{Out}(H)$ is cohomologous to a cocycle with values in a finite subgroup of $\mathrm{Out}(H)$. This provides a dynamical version of theorems of Farb--Kaimanovich--Masur and Bridson--Wade asserting that every morphism from $G$ to either the mapping class group of a finite-type surface or the outer automorphism group of a free group, has finite image. The main new geometric tool is a barycenter map that associates to every triple of points in the boundary of the (relative) free factor graph a finite set of (relative) free splittings.
