Boundedly simple groups of automorphisms of trees

TL;DR

This paper characterizes when groups of automorphisms of trees are boundedly simple, showing that only certain uniform subdivisions of biregular trees have this property, specifically 8-bounded simplicity.

Contribution

It proves that bounded simplicity occurs only for uniform subdivisions of biregular trees and establishes the exact boundedness level as 8, extending understanding of automorphism groups of trees.

Findings

Only uniform subdivisions of biregular trees are boundedly simple.

Such groups are exactly 8-boundedly simple.

Boundedly simple groups acting on trees have specific fixed or stabilizing properties.

Abstract

A group is boundedly simple if, for some constant N, every nontrivial conjugacy class generates the whole group in N steps. For a large class of trees, Tits proved simplicity of a canonical subgroup of the automorphism group, which is generated by pointwise stabilizers of edges. We prove that only for uniform subdivisions of biregular trees are such groups boundedly simple. In fact these groups are 8-boundedly simple. As a consequence, we prove that if G is boundedly simple (or from a certain class K) and G acts by automorphisms on a tree, then G fixes some vertex of A, or stabilizes some end of A, or the smallest nonempty G-invariant subtree of A is a uniform subdivision of a biregular tree.