In this second article, we continue to study classes of groups constructed from a functorial method due to Vaughan Jones. A key observation of the author shows that these groups admit remarkable diagrammatic descriptions that can be used to deduce their properties. Given any group and two of its endomorphisms, we construct a semi-direct product. In our first article dedicated to this construction, we classify up to isomorphism all these semi-direct products when one of the endomorphisms is trivial and describe their automorphism group. In this article, we focus on the case where both endomorphisms are automorphisms. The situation is rather different, and we obtain semi-direct products where the largest Richard Thompson’s group V is acting on some discrete analogues of loop groups. Note that these semi-direct products appear naturally in recent constructions of quantum field theories. Moreover, they were previously studied by Tanushevski and can be constructed via the framework of cloning systems of Witzel–Zaremsky. In particular, they provide examples of groups with various finiteness properties and possible counterexamples of a conjecture of Lehnert on co-context-free groups.
We provide a partial classification of these semi-direct products and describe their automorphism group explicitly. Moreover, we prove that groups studied in the first and second articles are never isomorphic to each other nor do they admit nice embeddings between them. We end the article with an appendix comparing Jones’ technology with Witzel–Zaremsky’s cloning systems and with Tanushevski’s construction. As in the first article, it was possible to achieve all the results presented via a surprising rigidity phenomenon on isomorphisms between these groups.
