Approximations of stable actions on $\mathbb{R}$-trees

Abstract

This article shows how to approximate a stable action of a finitely presented group on an $\mathbb{R}$ -tree by a simplicial one while keeping control over arc stabilizers. For instance, every small action of a hyperbolic group on an $\mathbb{R}$ -tree can be approximated by a small action of the same group on a simplicial tree. The techniques we use highly rely on Rips's study of stable actions on $\mathbb{R}$ -trees and on the dynamical study of exotic components by D. Gaboriau.