Relative Rips Machine

TL;DR

This paper introduces a relative Rips machine to analyze pairs of isometric group actions on R-trees, especially focusing on a group and its subgroup's actions on invariant subtrees, advancing understanding of their structure.

Contribution

The paper develops a novel relative Rips machine framework for studying pairs of R-tree actions, extending previous methods to include subgroup actions and invariant subtrees.

Findings

Provides a new tool for analyzing group actions on R-trees

Enables detailed study of subgroup actions on invariant subtrees

Enhances understanding of the structure of isometric group actions

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.