Finitely Presented Groups Acting on Trees

TL;DR

The paper demonstrates that finitely presented groups acting on -trees can be decomposed into graphs of groups, and non-trivial actions lead to actions on simplicial -trees, advancing understanding of group actions on trees.

Contribution

It establishes a decomposition of finitely presented groups acting on -trees into graphs of groups, linking group actions to tree structures.

Findings

Any action of a finitely presented group on an -tree can be decomposed into a graph of groups.

Non-trivial actions on -trees imply the existence of actions on simplicial -trees.

Provides a structural understanding of finitely presented groups acting on -trees.

Abstract

It is shown that for any action of a finitely presented group $G$ on an $\R$-tree, there is a decomposition of $G$ as the fundamental group of a graph of groups related to this action. If the action of $G$ on $T$ is non-trivial, i.e. there is no global fixed point, then $G$ has a non-trivial action on a simplcial $\R $-tree.