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Register a new userSubgroup decomposition in $text{Out}(F_n)$, Part IV: Relatively irreducible subgroups
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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.
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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 lamination pair, we study which lines and conjugacy classes in $F_n$ are weakly attracted to that lamination pair under forward and backward iteration respectively. For conjugacy classes, we prove Theorem F from the research annoucement, which exhibits a unique vertex group system called the nonattracting subgroup system having the property that the conjugacy classes it carries are characterized as those which are not weakly attracted to the attracting lamination under forward iteration, and also as those which are not weakly attracted to the repelling lamination under backward iteration. For lines in general, we prove Theorem G that characterizes exactly which lines are weakly attracted to the attracting lamination under forward iteration and which to the repelling lamination under backward iteration. We also prove Theorem H which gives a uniform version of weak attraction of lines. v3: Contains a stronger proof of Lemma 2.19 (part of the proof of Theorem G) for purposes of further applications.
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 purposes of constructing quasimorphisms. Here in Part I, after presenting the general theory, we focus on the case of infinite lamination subgroups $Gamma$ - those for which the set of all attracting laminations of all elements of $Gamma$ is infinite - using actions on free splitting complexes of free groups.
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 $ell^1$. Here in Part II we focus on finite lamination subgroups $Gamma$ --- meaning that the set of all attracting laminations of elements of $Gamma$ is finite --- and on the construction of hyperbolic actions of those subgroups to which the general theory of Part I is applicable.
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 whether or not there is an element $thetain mathsf{Out}(F_n)$ such that $theta(vec K)=vec K$ together with one such $theta$ if it exists and (2) a finite presentation for the subgroup of $mathsf{Out}(F_n)$ fixing $vec K$. S. Kalajdv{z}ievski published a verification of this algorithm. We present a different algorithm from the point of view of Culler-Vogtmanns Outer space. New results include that the subgroup of $mathsf{Out}(F_n)$ fixing $vec K$ is of type $mathsf{VF}$, an equivariant version of these results, an application, and a unified approach to such questions.
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