Von Neumann Algebras of Thompson-like Groups from Cloning Systems II

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).