Classification of Thompson Related Groups Arising from Jones Technology I

In the quest in constructing conformal field theories (CFTs), Jones has discovered a beautiful and deep connection between CFT, Richard Thompson’s groups, and knot theory. This led to a powerful functorial framework for constructing actions of particular groups arising from categories such as Thompson’s groups and braid groups. In particular, given a group and two of its endomorphisms one can construct a semidirect product where the largest Thompson’s group is acting. These semidirect products have remarkable diagrammatic descriptions that were previously used to provide new examples of groups having the Haagerup property. They naturally appear in certain field theories as being generated by local and global symmetries. Moreover, these groups occur in a construction of Tanushevski and can be realised using Brin–Zappa–Szep’s products with the technology of cloning systems of Witzel and Zaremsky. We consider in this article the class of groups obtained in that way where one of the endomorphism is trivial leaving the case of two nontrivial endomorphisms to a 2nd article. We provide an explicit description of all these groups as permutational restricted twisted wreath products where is the group acting and the twist depends on the endomorphism chosen. We classify this class of groups up to isomorphisms and provide a thin description of their automorphism group thanks to an unexpected rigidity phenomena.