Fully irreducible Automorphisms of the Free Group via Dehn twisting in $\sharp_k(S^2 \times S^1)$

TL;DR

This paper introduces a geometric Dehn twist approach in connected sums of spheres and circles to generate free groups of automorphisms of free groups, identifying conditions for fully irreducible and atoroidal automorphisms.

Contribution

It establishes a new method using geometric Dehn twists in $ atural_k(S^2 imes S^1)$ to produce fully irreducible automorphisms of free groups, linking geometric projections to algebraic properties.

Findings

Dehn twists generate free groups of automorphisms under certain conditions.

Elements are either conjugates of Dehn twists or fully irreducible automorphisms.

Produced automorphisms are atoroidal and fully irreducible when projected appropriately.

Abstract

By using a notion of a geometric Dehn twist in $\sharp_k(S^2 \times S^1)$, we prove that when projections of two $\mathbb{Z}$-splittings to the free factor complex are far enough from each other in the free factor complex, Dehn twist automorphisms corresponding to the $\mathbb{Z}$-splittings generate a free group of rank $2$. Moreover, every element from this free group is either conjugate to a power of one of the Dehn twists or it is a fully irreducible outer automorphism of the free group. We also prove that, when projected to the intersection graph, the same group of Dehn twists produce atoroidal fully irreducible automorphisms.