Relatively irreducible free subroups in Out($\mathbb{F}$)

Abstract: We prove that given a finite rank free group $\mathbb{F}$ of rank $\geq 3$ and two exponentially growing outer automorphisms $\psi$ and $\phi$ with dual lamination pairs $\Lambda^\pm_\psi$ and $\Lambda^\pm_\phi$ associated to them, and given a free factor system $\mathcal{F}$ with co-edge number $\geq 2$, $\phi, \psi $ each preserving $\mathcal{F}$, so that the pair $(\phi, \Lambda^\pm_\phi), (\psi, \Lambda^\pm_\psi)$ is independent relative to $\mathcal{F}$, then there $\exists$ $M\geq 1$, such that for any integer $m,n \geq M$, the group $\langle \phi^m, \psi^n \rangle$ is a free group of rank 2, all of whose non-trivial elements except perhaps the powers of $\phi, \psi$ and their conjugates, are fully irreducible relative to $\mathcal{F}$ with a lamination pair which fills relative to $\mathcal{F}$. In addition if both $\Lambda^\pm_\phi, \Lambda^\pm_\psi$ are non-geometric then this lamination pair is also non-geometric. We also prove that the extension groups induced by such subgroups will be relatively hyperbolic under some natural conditions.