Forest-skein categories and groups

Arnaud Brothier – 3 April 2024

Vaughan Jones found unexpected connections between subfactor theory and Richard Thompson's group while attempting to construct conformal field theories (CFT). This led to numerous fruitful applications and among others provided a novel way to construct group actions using categories. I am initiating a program strengthening Jones' visionary work where the Thompson group is replaced by a family of groups that I name "forest-skein groups". These groups are interesting on their own, satisfying exceptional properties and having powerful extra-structures for studying them. They are isotropy subgroup of forest-skein categories: categories of planar diagrams mod out by certain skein relations (just like planar algebras are in Jones' subfactor framework). I will briefly say a word about the story of Jones's discovery and then present forest-skein categories and groups using explicit examples.