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

Abstract

We prove that given a finite rank free group $\mathbb{F}$ of rank $\ge 3$ and two exponentially growing outer automorphisms $\psi$ and $\varphi$ with dual lamination pairs ${\Lambda }_{\psi }^{\pm }$ and ${\Lambda }_{\varphi }^{\pm }$ associated to them, and given a free factor system $\mathcal{F}$ with co-edge number $\ge 2$, $\varphi ,\psi$ each preserving $\mathcal{F}$, so that the pair $(\varphi ,{\Lambda }_{\varphi }^{\pm }),(\psi ,{\Lambda }_{\psi }^{\pm })$ is independent relative to $\mathcal{F}$, then there $\exists$ $M\ge 1$, such that for any integer $m,n\ge M$, the group $⟨{\varphi }^m,{\psi }^n⟩$ is a free group of rank 2, all of whose non-trivial elements except perhaps the powers of $\varphi ,\psi$ and their conjugates, are fully irreducible relative to $\mathcal{F}$ with a lamination pair which fills relative to $\mathcal{F}$. In addition if both ${\Lambda }_{\varphi }^{\pm },{\Lambda }_{\psi }^{\pm }$ 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.

Minor corrections, notation changes, updated references. Some proofs rewritten for better readability