Relative Rips Machine

Abstract

We study isometric actions of finitely presented groups on $\mathbb{R}$-trees. In this paper, we develop a relative version of the Rips machine to study $\textit{pairs}$ of such actions. An important example of a $\textit{pair}$ is a group action on an $\mathbb{R}$-tree and a subgroup action on its minimal invariant subtree.