论文标题
测量自由组和自由群体因素的等效嵌入
Measure equivalence embeddings of free groups and free group factors
论文作者
论文摘要
我们提供了一个简单明了的证据，表明免费组$ \ mathbb {f} _2 $允许嵌入到任何不可固定的本地紧凑型第二可计数（LCSC）组$ g $中的度量等效性。我们用它来证明，每个不合同的LCSC组$ g $都承认任何可能的Krieger类型的强烈狂热的动作，并承认具有任何规定的重量流动的不可否定的，弱混合的动作。我们还介绍了衡量等效性的概念，并针对$ ii_1 $因素进行了衡量等效性嵌入。我们证明，仅当自由组因子$ l（\ mathbb {f} _2）$时，$ ii_1 $因数$ m $是不合格的。我们证明了财产（T）的稳定性和在衡量$ II_1 $因素的量度下的Haagerup财产。
We give a simple and explicit proof that the free group $\mathbb{F}_2$ admits a measure equivalence embedding into any nonamenable locally compact second countable (lcsc) group $G$. We use this to prove that every nonamenable lcsc group $G$ admits strongly ergodic actions of any possible Krieger type and admits nonamenable, weakly mixing actions with any prescribed flow of weights. We also introduce concepts of measure equivalence and measure equivalence embeddings for $II_1$ factors. We prove that a $II_1$ factor $M$ is nonamenable if and only if the free group factor $L(\mathbb{F}_2)$ admits a measure equivalence embedding into $M$. We prove stability of property (T) and the Haagerup property under measure equivalence of $II_1$ factors.
