Fully irreducible automorphisms of the free group via Dehn twisting in $\sharp_k(S^2 \times S^1)$

Abstract

By using a notion of a geometric Dehn twist in ${♯}_k\left(S^2\times S^1\right)$, 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 either is conjugate to a power of one of the Dehn twists or is a fully irreducible outer automorphism of the free group. We also prove that, when the projections of $\mathbb{Z}$–splittings are sufficiently far away from each other in the intersection graph, the group generated by the Dehn twists has automorphisms that are either conjugate to Dehn twists or atoroidal fully irreducible.