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

TL;DR

This paper investigates the behavior of Dehn twists within the outer automorphism group of free groups, revealing conditions under which their powers generate right-angled Artin groups through geometric analysis.

Contribution

It introduces new criteria for Dehn twists to generate right-angled Artin groups and constructs a space where these twists act parabolically, advancing understanding of their dynamics.

Findings

Large powers of Dehn twists can generate right-angled Artin groups.

Established a criterion for commuting Dehn twists.

Constructed a space with parabolic action of Dehn twists.

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.