2021/04/10 by Eli Bashwinger, Matthew C. B. Zaremsky, Bashwinger, Eli +1 · 2 citations
Computer Science · Mathematics · #20F65 #46L10 #Advanced Operator Algebra Research #FOS: Mathematics #Geometric and Algebraic Topology #Group Theory (math.GR) #Operator Algebras (math.OA) #semigroups and automata theory
paper · pdf · doi:10.48550/arxiv.2104.04826
openalex publication_date 2021/04/10 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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 (Gn)n∈ℕ with a d-ary cloning system, we get a Thompson-like group \mathscrTd(G_*), and in this paper we find some mild, natural conditions under which the group von Neumann algebra L(\mathscrTd(G_*)) has desirable properties. For instance, if the d-ary cloning system is "fully compatible" and "diverse" then we prove that L(\mathscrTd(G_*)) is a type \textrmII1 factor. If moreover the d-ary cloning system is "uniform" and "slightly pure" then we prove L(\mathscrTd(G_*)) is even a McDuff factor, so \mathscrTd(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, L(bV) and L(bF) are type \textrmII1 factors and L(bF) is McDuff. In particular we get the surprising result that bF is inner amenable.