Normal subgroups of groups acting on trees and automorphism groups of graphs

TL;DR

This paper investigates the structure of certain subgroups of automorphism groups of trees, showing that under specific conditions, the subgroup generated by stabilizers of half-trees is topologically simple, with applications to graph automorphisms.

Contribution

It establishes the topological simplicity of subgroups generated by half-tree stabilizers in automorphism groups of trees, extending previous results and applying to graph automorphisms.

Findings

Subgroups generated by half-tree stabilizers are topologically simple.

Results extend recent work of Caprace and De Medts (2011).

Applications to automorphism groups of locally finite primitive graphs with infinitely many ends.

Abstract

Let $T$ be a tree and $e$ an edge in $T$. If $C$ is a component of $T\setminus e$ and both $C$ and its complement are infinite we say that $C$ is a half-tree. The main result of this paper is that if $G$ is a closed subgroup of the automorphism group of $T$ and $G$ leaves no non-trivial subtree invariant and fixes no end of $T$ then the subgroup generated by the pointwise stabilizers of half-trees is topologically simple. This result is used to derive analogues of recent results of Caprace and De Medts (2011) and it is also applied in the study of the full automorphism group of a locally finite primitive graph with infinitely many ends.