Forest-skein groups III: simplicity

TL;DR

This paper explores the simplicity of forest-skein groups, providing characterizations and constructing examples linked to binary trees and Higman-Thompson groups, advancing understanding of their algebraic structure.

Contribution

It introduces dynamical and categorical characterizations of simple derived subgroups in forest-skein groups and constructs new classes of simple groups from binary trees and Higman-Thompson groups.

Findings

Characterizations of when forest-skein groups have simple derived subgroups

Construction of simple groups from binary trees

Construction of simple groups from n-ary Higman-Thompson groups

Abstract

An Ore forest-skein category provides three forest-skein groups equipped with a powerful diagrammatic calculus analogous to Richard Thompson's groups F,T,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 classes of examples. The first associates two finitely presented simple groups to every finite binary tree and the second associates two simple groups to every n-ary Higman-Thompson group.