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

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