On Whitehead's cut vertex lemma

TL;DR

This paper extends Whitehead's cut vertex lemma to conjugacy classes and convex-cocompact subgroups in groups acting cocompactly on trees, providing a broader understanding of the lemma's applicability.

Contribution

It generalizes Whitehead's lemma from free groups to groups acting on trees with finitely generated edge stabilizers, including conjugacy classes and subgroups.

Findings

Proves a version of Whitehead's lemma for conjugacy classes.

Establishes the lemma for convex-cocompact subgroups.

Provides structural insights into groups acting on trees.

Abstract

One version of Whitehead's famous cut vertex lemma says that if an element of a free group is part of a free basis, then a certain graph associated to its conjugacy class that we call the star graph is either disconnected or has a cut vertex. We state and prove a version of this lemma for conjugacy classes of elements and convex-cocompact subgroups of groups acting cocompactly on trees with finitely generated edge stabilizers.