  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.
