源自克隆系统的类Thompson群的von Neumann代数 II
Von Neumann Algebras of Thompson-like Groups from Cloning Systems II
AI总结：
本文进一步分析了源自克隆系统的类Thompson群von Neumann代数结构，证明了在“多样”克隆系统下包含关系的弱渐近同态性质、不可约性与奇异性，并给出非多样系统反例及因子包含与McDuff性质的应用。
摘要：
设 $(G_n)_{n \in \mathbb{N}}$ 为配备 $d$ 元克隆系统的群序列，并记 $\mathscr{T}_d(G_*)$ 为所得的类 Thompson 群。在与 Zaremsky 合作的前期工作中，我们获得了关于 $\mathscr{T}_d(G_*)$ 的群 von Neumann 代数（记为 $L(\mathscr{T}_d(G_*))$）的结构结果。在对 $d$ 元克隆系统的一些自然假设下，我们证明了 $L(\mathscr{T}_d(G_*))$ 是 $\text{II}_1$ 型因子。在增加几个自然假设后，我们进一步证明了 $L(\mathscr{T}_d(G_*))$ 还是 McDuff 因子。在本文中，我们进一步分析了 $L(\mathscr{T}_d(G_*))$ 的结构，特别是包含关系 $L(F_d) \subseteq L(\mathscr{T}_d(G_*))$，其中 $F_d$ 是最小的 Higman--Thompson 群。我们证明，如果 $d$ 元克隆系统是“多样的”，那么 $L(F_d) \subseteq L(\mathscr{T}_d(G_*))$ 满足弱渐近同态性质。由此可得，该包含关系是不可约的，这是对我们关于 $L(\mathscr{T}_d(G_*))$ 是 $\text{II}_1$ 型因子结果的重大改进，且该包含关系也是奇异的。随后，我们考察了关于弱渐近同态性质、奇异性及不可约性的非多样 $d$ 元克隆系统的例子。最后，我们以一些应用结束本文。我们构造了一个机器，它接收任意群和有限群，并生成 $\text{II}_1$ 型因子的包含关系（有限和无限指标均有），这些包含关系是奇异的但不满足弱渐近同态性质。最后，利用包含关系 $L(F_d) \subseteq L(\mathscr{T}_d(G_*))$ 的不可约性、我们关于 $L(\mathscr{T}_d(G_*))$ 何时为 McDuff 因子的条件，以及 Higman-Thompson 群 $F_d$ 是特征刚性（Peterson 意义下）的事实，我们证明了群 $F_d$ 是 McDuff（Deprez-Vaes 意义下）。
英文摘要：
Let $(G_n)_{n \in \mathbb{N}}$ be a sequence of groups equipped with a $d$-ary cloning system and denote by $\mathscr{T}_d(G_*)$ the resulting Thompson-like group. In previous work joint with Zaremsky, we obtained structural results concerning the group von Neumann algebra of $\mathscr{T}_d(G_*)$, denoted by $L(\mathscr{T}_d(G_*))$. Under some natural assumptions on the $d$-ary cloning system, we proved that $L(\mathscr{T}_d(G_*))$ is a type $\text{II}_1$ factor. With a few additional natural assumptions, we proved that $L(\mathscr{T}_d(G_*))$ is, moreover, a McDuff factor. In this paper, we further analyze the structure of $L(\mathscr{T}_d(G_*))$, in particular the inclusion $L(F_d) \subseteq L(\mathscr{T}_d(G_*))$, where $F_d$ is the smallest of the Higman--Thompson groups. We prove that if the $d$-ary cloning system is ``diverse," then $L(F_d) \subseteq L(\mathscr{T}_d(G_*))$ satisfies the weak asymptotic homomorphism property. As a consequence, the inclusion is irreducible, which is a considerable improvement of our result that $L(\mathscr{T}_d(G_*))$ is a type $\text{II}_1$ factor, and the inclusion is also singular. Then we look at examples of non-diverse $d$-ary cloning systems with respect to the weak asymptotic homomorphism property, singularity, and irreducibility. Then we finish the paper with some applications. We construct a machine which takes in an arbitrary group and finite group and produces an inclusion (both finite and infinite index) of type $\text{II}_1$ factors which is singular but without the weak asymptotic homomorphism property. Finally, using irreducibility of the inclusion $L(F_d) \subseteq L(\mathscr{T}_d(G_*))$, our conditions for when $L(\mathscr{T}_d(G_*))$ is a McDuff factor, and the fact that Higman-Thompson groups $F_d$ are character rigid (in the sense of Peterson), we prove that the groups $F_d$ are McDuff (in the sense of Deprez-Vaes).
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