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.
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