论文标题
边界的不合适性和测量二维ARTIN双曲线类型的等效性刚度
Boundary amenability and measure equivalence rigidity among two-dimensional Artin groups of hyperbolic type
论文作者
论文摘要
我们从测量等效性的角度研究$ 2 $维二维的ARTIN ARTIN组，并建立刚性定理。 我们首先证明它们是边界的。因此，每个群体都会通过简单的异构体在连接的双曲线$ \ mathrm {cat}（ - 1）$简单复合物上分散作用，并且在有限的许多等轴测类型中，有很多简单的复合物，假设顶点稳定剂是有限的许多等轴测图中的。因此，他们满足了诺维科夫的猜想。 然后，我们证明了等效$ 2 $ 2 $维的ARTIN ARTIN组的双曲线类型具有同构固定集图 - Crisp引入的曲线图的类似图。这会产生分类结果。 我们获得强刚性定理。令$ g =g_γ$为$ 2 $ - 维二曲类型类型，带有$ \ mathrm {out}（g）$有限。当固定集图和Cayley Complex $ \ Mathfrak {C} $的自动形态组相吻合时，每个可计数的组$ h $，相当于$ g $，与$ \ mathrm {autrm {aut}（autfrak}（\ mathfrak {c}）的lattice相当于$ g $。每当$γ$不含三角形，所有标签至少$ 3 $时，都会发生这种情况 - 除非$ g $可与$ \ mathbb {z} $的直接总和和一个免费组相称。当$γ$满足额外的星形依据条件时，$ \ mathrm {aut}（\ mathfrak {c}）$是可计数的，而$ h $几乎是$ g $的同构。 这在轨道等效性刚度上有应用，并且与Artin组的千古动作相关的von Neumann代数的刚度结果。我们还获得了有关某些Artin组可能的晶格信封的刚性声明，以及从高级晶格到$ 2 $维二维的双曲线类型的Cocycle Superrigities定理。
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.
