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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.
This is the third 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, given an outer automorphism of $F_n$ and an attracting-repelling la
In this two part work we prove that for every finitely generated subgroup $Gamma < text{Out}(F_n)$, either $Gamma$ is virtually abelian or $H^2_b(Gamma;mathbb{R})$ contains an embedding of $ell^1$. The method uses actions on hyperbolic spaces, for pu
This is the second part of a two part work in which we prove that for every finitely generated subgroup $Gamma < mathsf{Out}(F_n)$, either $Gamma$ is virtually abelian or its second bounded cohomology $H^2_b(Gamma;mathbb{R})$ contains an embedding of
We give a short proof of Masbaum and Reids result that mapping class groups involve any finite group, appealing to free quotients of surface groups and a result of Gilman, following Dunfield-Thurston.
S. Gersten announced an algorithm that takes as input two finite sequences $vec K=(K_1,dots, K_N)$ and $vec K=(K_1,dots, K_N)$ of conjugacy classes of finitely generated subgroups of $F_n$ and outputs: (1) $mathsf{YES}$ or $mathsf{NO}$ depending on