$\mathbb{R}$--trees and accessibility over arc-stabilisers

TL;DR

This paper characterizes point-stabilizers in minimal actions of finitely presented groups on $ ext{R}$-trees, linking them to simplicial trees and exploring implications for automorphisms of right-angled Artin groups.

Contribution

It provides a description of point-stabilizers in $ ext{R}$-trees under accessibility assumptions, showing they are finitely generated and related to simplicial trees.

Findings

Point-stabilizers are finitely generated.

Description of point-stabilizers in terms of simplicial trees.

Applications to automorphisms of right-angled Artin groups.

Abstract

Let $G\curvearrowright T$ be a minimal action on an $\mathbb{R}$--tree with $G$ finitely presented. Assuming that $G$ is accessible over the family of arc-stabilisers of $T$, we give a description of the point-stabilisers of $T$ in terms of simplicial trees. In particular, these point-stabilisers are finitely generated. This has applications to the study of automorphisms of right-angled Artin groups and special groups.