Von Neumann algebras of Thompson-like groups from cloning systems

Abstract

We prove a variety of results about the group von Neumann algebras associated to Thompson-like groups arising from so called $d$-ary cloning systems. Cloning systems are a framework developed by Witzel and the second author, with a $d$-ary version subsequently developed by Skipper and the second author, which can be used to construct generalizations of the classical Thompson's groups $F$, $T$, and $V$. Given a family of groups $(G_n)_{n\in\mathbb{N}}$ with a $d$-ary cloning system, we get a Thompson-like group $\mathscr{T}_d(G_*)$, and in this paper we find some mild, natural conditions under which the group von Neumann algebra $\mathcal{L}(\mathscr{T}_d(G_*))$ has desirable properties. For instance, if the $d$-ary cloning system is "fully compatible" and "diverse" then we prove that $\mathcal{L}(\mathscr{T}_d(G_*))$ is a type $\textrm{II}_1$ factor. If moreover the $d$-ary cloning system is "uniform" and "slightly pure" then we prove $\mathcal{L}(\mathscr{T}_d(G_*))$ is even a McDuff factor, so $\mathscr{T}_d(G_*)$ is inner amenable. Examples of $d$-ary cloning systems satisfying these conditions are easy to come by, and include many existing examples, for instance our results show that for $bV$ and $bF$ the Brin-Dehornoy braided Thompson group and pure braided Thompson group, $\mathcal{L}(bV)$ and $\mathcal{L}(bF)$ are type $\textrm{II}_1$ factors and $\mathcal{L}(bF)$ is McDuff. In particular we get the surprising result that $bF$ is inner amenable.