Boundedly Simple Groups of Automorphisms of Trees

A group is boundedly simple if for some constant N , every nontrivial conjugacy class generates the whole group in N steps (bounded simplicity implies simplicity). For a large class of trees, Tits proved simplicity of automorphism groups generated by stabilisers of edges. We determine the subclass of trees for which such groups are boundedly simple. This is the class of subdivisions of bi-regular trees and related groups are 32-boundedly simple. As a consequence, we show that if a boundedly simple group G acts by automorphisms on a tree and there is nontrivial stabiliser of some edge in G, then there is G-invariant subtree which is a subdivision of a bi-regular tree.