论文标题
通过可弹性变形测量等效性刚度
Measure equivalence rigidity via s-malleable deformations
论文作者
论文摘要
我们挑选一大类$ {\ MATHSCR {m}} $，以下唯一的质量分解结果：如果$γ_1，\ dots，γ_n\ in {\ Mathscr {m}}} $ and $γ_1和$γ_1\ times \ times \ times \ times \ times pimec_n $无限ICC组，然后是$ n \ ge m $，如果$ n = m $，则在索引排列后，$γ_i$的量度相当于$λ_i$，对于所有$ 1 \ leq i \ leq n $。这为属于$ {\ mathscr {m}} $的组提供了monod和shalom定理\ cite {ms02}的类似物。类$ {\ Mathscr {m}} $是使用von Neumann代数的组构建的，它承认从Sorin Popa的意义上承认可示意的变形，并且其中包含所有ICC $γ$的$γ$（i）$γ$的$γ$，其$γ$是与正常的$γ$ coce coce coce coce coce to n of comploble comploable of comploble coce coce coce coce coce nimpclociation。因此，我们为属于$ {\ mathscr {m}} $的产品组的动作得出了几个轨道等效刚度结果。最后，对于$γ$满足条件（II）的组，我们表明，von Neumann组的所有嵌入是不可违反的内部不可木的组成$ l（γ）$的所有嵌入，均为``jirdig''。我们特别为Popa提供了一种替代性解决方案。
We single out a large class of groups ${\mathscr{M}}$ for which the following unique prime factorization result holds: if $Γ_1,\dots,Γ_n\in {\mathscr{M}}$ and $Γ_1\times\dots\timesΓ_n$ is measure equivalent to a product $Λ_1\times\dots\timesΛ_m$ of infinite icc groups, then $n \ge m$, and if $n = m$ then, after permutation of the indices, $Γ_i$ is measure equivalent to $Λ_i$, for all $1\leq i\leq n$. This provides an analogue of Monod and Shalom's theorem \cite{MS02} for groups that belong to ${\mathscr{M}}$. Class ${\mathscr{M}}$ is constructed using groups whose von Neumann algebras admit an s-malleable deformation in the sense of Sorin Popa and it contains all icc non-amenable groups $Γ$ for which either (i) $Γ$ is an arbitrary wreath product group with amenable base or (ii) $Γ$ admits an unbounded 1-cocycle into its left regular representation. Consequently, we derive several orbit equivalence rigidity results for actions of product groups that belong to ${\mathscr{M}}$. Finally, for groups $Γ$ satisfying condition (ii), we show that all embeddings of group von Neumann algebras of non-amenable inner amenable groups into $L(Γ)$ are ``rigid". In particular, we provide an alternative solution to a question of Popa that was recently answered in \cite{DKEP22}.
