Rigidity and automorphisms of groups constructed using Jones' technology

Jones' technology, developed by Vaughan Jones during his exploration of the connections between conformal field theory and subfactors, is a powerful mechanism for generating actions of groups coming from categories, notably Richard Thompson's groups $F \subseteq T \subseteq V$. We give a structure theorem for the isomorphisms between split extensions of Thompson's group $V$ arising from Jones' technology, generalising results of Brothier. Using this structure theorem, we classify a family of unrestricted, twisted permutational wreath products up to isomorphism, and decompose their automorphism groups in the untwisted case. These unrestricted wreath products arise from applying Jones' technology to contravariant monoidal functors. In contrast, using covariant functors, Brothier constructed a large class of restricted wreath products, classified them up to isomorphism, and completely described their automorphism groups. Our work broadens Brothier's findings and highlights the duality between groups constructed using covariant and contravariant functors.