Classification of Thompson related groups arising from Jones technology I

Abstract

In the quest in constructing conformal field theories (CFT) Jones has discovered a beautiful and deep connection between CFT, Richard Thompson's groups and knot theory. This led to a powerful framework for constructing actions of particular groups arising from categories such as Richard Thompson's groups and braid groups. In particular, given a group and two of its endomorphisms one can construct a semidirect product. Those semidirect products have remarkable diagrammatic descriptions which were previously used to provide new examples of groups having the Haagerup property. Moreover, they naturally appear in certain field theories as being generated by local and global symmetries.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 second article. The groups obtained can be described in terms of permutational restricted twisted wreath products containing the larger Thompson group. We classify this class of groups up to isomorphisms and provide a thin description of their automorphism group thanks to an unexpected rigidity phenomena.À la mémoire de Vaughan Jones