[论文解读] Boundary amenability and measure equivalence rigidity among two-dimensional Artin groups of hyperbolic type

Camille Horbez, Jingyin Huang|arXiv (Cornell University)|Apr 20, 2020

Advanced Operator Algebra Research参考文献 88被引用 5

一句话总结

该论文建立了双曲型二维Artin群的边界拟平凡性与测度等价刚性，证明其满足诺维科夫猜想，并在测度等价下具有强刚性。通过在CAT(−1)复形上的作用及固定点图，证明测度等价的群与Cayley复形自同构群的格子共轭，当定义图无三角形且标签≥3时，刚性更强。

ABSTRACT

We study $2$-dimensional Artin groups of hyperbolic type from the viewpoint of measure equivalence, and establish rigidity theorems. We first prove that they are boundary amenable. So is every group acting discretely by simplicial isometries on a connected piecewise hyperbolic $\mathrm{CAT}(-1)$ simplicial complex with countably many simplices in finitely many isometry types, assuming that vertex stabilizers are boundary amenable. Consequently, they satisfy the Novikov conjecture. We then show that measure equivalent $2$-dimensional Artin groups of hyperbolic type have isomorphic fixed set graphs -- an analogue of the curve graph, introduced by Crisp. This yields classification results. We obtain strong rigidity theorems. Let $G=G_Γ$ be a $2$-dimensional Artin group of hyperbolic type, with $\mathrm{Out}(G)$ finite. When the automorphism groups of the fixed set graph and of the Cayley complex $\mathfrak{C}$ coincide, every countable group $H$ which is measure equivalent to $G$, is commensurable to a lattice in $\mathrm{Aut}(\mathfrak{C})$. This happens whenever $Γ$ is triangle-free with all labels at least $3$ -- unless $G$ is commensurable to the direct sum of $\mathbb{Z}$ and a free group. When $Γ$ satisfies an additional star-rigidity condition, then $\mathrm{Aut}(\mathfrak{C})$ is countable, and $H$ is almost isomorphic to $G$. This has applications to orbit equivalence rigidity, and rigidity results for von Neumann algebras associated to ergodic actions of Artin groups. We also derive a rigidity statement regarding possible lattice envelopes of certain Artin groups, and a cocycle superrigidity theorem from higher-rank lattices to $2$-dimensional Artin groups of hyperbolic type.

研究动机与目标

• 建立双曲型二维Artin群及其在CAT(−1)单纯复形上作用的推广群的边界拟平凡性。
• 证明这些群的测度等价刚性定理，表明测度等价的群与Cayley复形自同构群的格子共轭。
• 在附加的星形刚性或无三角形图条件下，推导强刚性结果，导致测度等价群几乎同构。
• 将结果应用于轨道等价刚性和与遍历作用相关的冯诺依曼代数的W*-刚性。
• 从高秩格子到这些Artin群建立上链超刚性定理，表明所有上链均同调于平凡上链。

提出的方法

• 证明群在可数、分段双曲的CAT(−1)单纯复形上离散作用的边界拟平凡性的一般判据，且单形的等距类型有限。
• 证明若顶点稳定子群为边界拟平凡，则群本身为边界拟平凡；若边稳定子群为拟平凡，则作用在视觉边界上为Borel拟平凡。
• 引入固定点图ΘΓ作为Artin群的曲线图类对象，推广曲面群的曲线图。
• 利用固定点图相对于抛物子群的几何刚性，通过Guelman-Hurtado-Lyndon的结果推导测度等价刚性。
• 应用高秩格子的上链刚性理论，证明从此类格子到Artin群的每个上链均同调于平凡上链。
• 利用固定点图中顶点稳定子群在虚拟意义下同构于F×Z（F为自由群）的事实，结合自由群与Z的已知上链刚性，将结果推广至乘积群。

实验结果

研究问题

• RQ1 在何种条件下，双曲型二维Artin群为边界拟平凡？
• RQ2 何时两个测度等价的双曲型二维Artin群共轭于Cayley复形自同构群的格子？
• RQ3 定义图需满足何种条件，才能使测度等价群几乎同构？
• RQ4 固定点图的结构如何与Artin群的测度等价性和刚性相关？
• RQ5 能否为映射到双曲型二维Artin群的高秩格子建立上链刚性定理？

主要发现

• 所有双曲型二维Artin群均为边界拟平凡，因此满足高阶示性类的诺维科夫猜想。
• 若定义图无三角形且所有标签≥3，且Out(G)有限，则每个与G测度等价的群均共轭于Aut(𝒞)（即Cayley复形自同构群）的格子。
• 当定义图满足星形刚性条件，且Aut(𝒞)可数时，每个测度等价群H均与G几乎同构。
• 固定点图ΘΓ为曲线图类对象，GΓ在其上作用几何刚性，从而可应用上链刚性工具。
• 从高秩格子或代数群到双曲型二维Artin群的每个上链均同调于平凡上链。
• 结果表明，与这些Artin群的遍历作用相关的冯诺依曼代数具有强W*-刚性，并对局部紧第二可数群中的可能格子包络提供了分类。