Forest-skein groups III: simplicity

Abstract

An Ore forest-skein category provides three forest-skein groups equipped with a powerful diagrammatic calculus analogous to Richard Thompson’s groups $F\subset T\subset V$. We investigate when forest-skein groups have simple derived subgroups and establish two characterisations: a dynamical one and a categorical one. We then construct two explicit families of examples. The first family associates a pair of finitely presented simple groups to every finite binary tree, and the second family extends in a new way the construction of Higman–Thompson groups, yielding a pair of simple groups for each natural number.