论文标题
测量希格曼组之间的等效刚度
Measure equivalence rigidity among the Higman groups
论文作者
论文摘要
我们证明，至少$ 5 $发电机的所有（广义）Higman组都是超级衡量等效性的。更准确地说，让$ k \ ge 5 $，让$ h $是一个带有发电机$ a_1，\ dots，a_k $和baumslag-solartiat的集团，由$ a_ia_ {i+1}^{m_i}^{m_i}^{m_i} a_i} a_i^a_i^{-1} = a_i^a_i^a_i^a_i^a_i^{n_i^{ $ \ mathbb {z}/k \ mathbb {z} $和nonzero Integers $ | m_i | \ neq | n_i | $用于每个$ i $。我们证明，每个相当于$ h $的可数群体实际上都是同构至$ h $的。证明中的一个关键成分是一般性语句，为在$ \ mathrm {cat}（-1）$ polyhedral confffectes上进行酰基辅助作用的组提供了测量的组理论不变性，并在顶点和边缘稳定器上控制。 在我们的工作的后果中，我们获得了有关其Cayley图的晶格嵌入和自动形态的广义希格曼组的刚性定理。我们还得出了所有自由，千古，概率测量的概率衡量措施的轨道等效性和$ w^*$ - 超级繁殖定理。
We prove that all (generalized) Higman groups on at least $5$ generators are superrigid for measure equivalence. More precisely, let $k\ge 5$, and let $H$ be a group with generators $a_1,\dots,a_k$, and Baumslag-Solitar relations given by $a_ia_{i+1}^{m_i}a_i^{-1}=a_i^{n_i}$, with $i$ varying in $\mathbb{Z}/k\mathbb{Z}$ and nonzero integers $|m_i|\neq |n_i|$ for each $i$. We prove that every countable group which is measure equivalent to $H$, is in fact virtually isomorphic to $H$. A key ingredient in the proof is a general statement providing measured group theoretic invariants for groups acting acylindrically on $\mathrm{CAT}(-1)$ polyhedral complexes with control on vertex and edge stabilizers. Among consequences of our work, we obtain rigidity theorems for generalized Higman groups with respect to lattice embeddings and automorphisms of their Cayley graphs. We also derive an orbit equivalence and $W^*$-superrigidity theorem for all free, ergodic, probability measure-preserving actions of generalized Higman groups.
