On Whitehead's cut vertex lemma

TL;DR

The paper extends Whitehead's cut vertex lemma to conjugacy classes of elements and convex-cocompact subgroups in groups acting cocompactly on trees with finitely generated edge stabilizers, by developing a framework of Stallings graphs of groups and almost $\mathbb{G}$ graphs of groups. Using Bassian morphisms, direction and turn dynamics, and a fold-based analysis in the reduced deformation space, it reduces the problem to the star graph associated with a Stallings graph of groups. The main result shows that if a collection of conjugacy classes is jointly simple, then the star graph is either disconnected or contains a cut vertex, providing a combinatorial criterion for simplicity within this setting. The approach blends ideas from Heusener–Weidmann and Vogtmann–Bestvina–Feighn–Handel, adapted to graphs of groups and deformation spaces, yielding a constructive method to detect decomposition phenomena via star graphs. The findings have potential implications for understanding the structure of convex-cocompact subgroups and conjugacy classes in groups acting on trees and graphs of groups.

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.

Figures (2)

• Figure 1: Right: The graph of groups $\mathbb{G}$. Left: The immersion $f\colon \mathcal{G} \to \mathbb{G}$.
• Figure 2: The star graph of the immersion $f\colon \mathcal{G} \to \mathbb{G}$.

Theorems & Definitions (9)

• Theorem A
• Example 1
• Example 2
• Lemma 3
• proof
• Proposition 4
• proof
• Example 5
• proof : Proof of \ref{['maintheorem']}