Subgroup decomposition in $\text{Out}(F_n)$, Part IV: Relatively irreducible subgroups

This is the fourth and last in a series of four papers (with research announcement posted on this arXiv) that develop a decomposition theory for subgroups of $\text{Out}(F_n)$. In this paper we develop general ping-pong techniques for the action of $\text{Out}(F_n)$ on the space of lines of $F_n$. Using these techniques we prove the main results stated in the research announcement, Theorem C and its special case Theorem I, the latter of which says that for any finitely generated subgroup $\mathcal H$ of $\text{Out}(F_n)$ that acts trivially on homology with $\mathbb{Z}/3$ coefficients, and for any free factor system $\mathcal F$ that does not consist of (the conjugacy classes of) a complementary pair of free factors of $F_n$ nor of a rank $n-1$ free factor, if $\mathcal H$ is fully irreducible relative to $\mathcal F$ then $\mathcal H$ has an element that is fully irreducible relative to $\mathcal F$. We also prove Theorem J which, under the additional hypothesis that $\mathcal H$ is geometric relative to $\mathcal F$, describes a strong relation between $\mathcal H$ and a mapping class group of a surface. v3 and 4: Strengthened statements of the main theorems, highlighting the role of the finite generation hypothesis, and providing an alternative hypothesis. Strengthened proofs of lamination ping-pong, and a strengthened conclusion in Theorem J, for further applications.