Dynamics of Dehn Twists in the Outer Automorphism Group of a Free Group

TL;DR

The paper addresses embedding right-angled Artin groups into the outer automorphism group of a free group by using Dehn twists along one-edge $\mathbb{Z}$-splittings. It transfers the problem to the mapping class group of the doubled handlebody and develops a robust geometric framework, including the bigon–bihedron criterion and a commuting criterion, together with a ping-pong style argument based on fellow-traveling curves and torus cores. The main contribution is an explicit injectivity criterion: for a compatible collection with coincidence graph $\Gamma$, sufficiently large powers $|n_i|$ yield an embedding $\Phi:A(\Gamma) \to \mathrm{Out}(\mathbb{F})$ with $T_i\mapsto \delta_i^{n_i}$, producing RAAGs generated by Dehn twists. Additionally, the authors construct a compact space on which Dehn twists act parabolically and develop a space of factorial curves to study asymptotic dynamics, linking combinatorial group theory with geometric topology in Out($\mathbb{F}$).

Abstract

We study Dehn twists in the outer automorphism group of a finitely generated non-abelian free group. Our main result states that, under certain compatibility conditions, sufficiently large powers of finitely many Dehn twists generate a right-angled Artin group. The proof proceeds by analyzing the geometry of spheres, tori, and simple closed curves in a doubled handlebody. Along the way, we establish the bigon--bihedron criterion and an equivalent condition for commuting Dehn twists. Furthermore, we construct a compact topological space on which every Dehn twist acts parabolically.

Theorems & Definitions (53)

• Theorem 1.1: Theorem \ref{['thm:main']}
• Theorem 1.2: Theorem \ref{['thm:commuting']}
• Proposition 1.3: Bigon--bihedron criterion, Proposition \ref{['prop:bigon-bihedron']}
• Theorem 1.4: Theorem \ref{['thm:converge']}
• Lemma 2.1
• proof
• Lemma 2.2
• proof
• Lemma 2.3
• proof