弱等价类的乘法可以是不连续的
Multiplication of Weak Equivalence Classes May Be Discontinuous
摘要：
对于可数无限群$Γ$，设$\mathcal{W}_Γ$表示无原子标准概率空间上的$Γ$的保测作用的所有弱等价类的空间，该空间具有由Abért和Elek引入的紧可度量化拓扑.在$\mathcal{W}_Γ$上有一个自然的乘法运算（由取作用的乘积引起），使$\mathcal{W}_Γ$成为阿贝尔半群。Burton、Kechris和Tamuz证明了如果$Γ$是顺从的，那么$\mathcal{W}_Γ$是拓扑半群，即，乘积映射$\mathcal{W}_Γ\times \mathcal{W}_Γ\to \mathcal{W}_Γ\colon (\mathfrak{a}, \mathfrak{b}) \mapsto \mathfrak{a} \times \mathfrak{b}$是连续的。与此相反，我们证明了，如果$Γ$是一个Zaplaki稠密子群的$\mathrm{SL}_d(\mathbb{Z})$的一些$d \geqslant 2$（例如，如果$Γ$是一个非阿贝尔自由群），那么在$\mathcal{W}_Γ$乘法是不连续的，即使是限制在子空间$\mathcal{FW}_Γ$的所有自由弱等价类。
英文摘要：
For a countably infinite group $Γ$, let $\mathcal{W}_Γ$ denote the space of all weak equivalence classes of measure-preserving actions of $Γ$ on atomless standard probability spaces, equipped with the compact metrizable topology introduced by Abért and Elek. There is a natural multiplication operation on $\mathcal{W}_Γ$ (induced by taking products of actions) that makes $\mathcal{W}_Γ$ an Abelian semigroup. Burton, Kechris, and Tamuz showed that if $Γ$ is amenable, then $\mathcal{W}_Γ$ is a topological semigroup, i.e., the product map $\mathcal{W}_Γ\times \mathcal{W}_Γ\to \mathcal{W}_Γ\colon (\mathfrak{a}, \mathfrak{b}) \mapsto \mathfrak{a} \times \mathfrak{b}$ is continuous. In contrast to that, we prove that if $Γ$ is a Zariski dense subgroup of $\mathrm{SL}_d(\mathbb{Z})$ for some $d \geqslant 2$ (for instance, if $Γ$ is a non-Abelian free group), then multiplication on $\mathcal{W}_Γ$ is discontinuous, even when restricted to the subspace $\mathcal{FW}_Γ$ of all free weak equivalence classes.
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