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

Abstract

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.

32 pages. Contains ross references to other parts of this series. All other parts of this series, including the research announcement, are found on this arXiv