Actions of finitely generated groups on R-trees

TL;DR

This paper investigates how finitely generated groups act on real trees, showing they either split over certain subgroups or can be constructed from basic actions, extending prior results with new counterexamples and a generalized Scott's Lemma.

Contribution

It extends existing theorems on group actions on R-trees by correcting previous statements and introducing a broader splitting criterion based on direct limits.

Findings

Groups either split over controlled subgroups or are built from basic actions.

Counterexamples demonstrate previous theorems were misstated.

A generalized Scott's Lemma is established for group splittings.

Abstract

We study actions of finitely generated groups on $\bbR$-trees under some stability hypotheses. We prove that either the group splits over some controlled subgroup (fixing an arc in particular), or the action can be obtained by gluing together actions of simple types: actions on simplicial trees, actions on lines, and actions coming from measured foliations on 2-orbifolds. This extends results by Sela and Rips-Sela. However, their results are misstated, and we give a counterexample to their statements. The proof relies on an extended version of Scott's Lemma of independent interest. This statement claims that if a group $G$ is a direct limit of groups having suitably compatible splittings, then $G$ splits.