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Register a new userHyperbolic actions and 2nd bounded cohomology of subgroups of $mathsf{Out}(F_n)$. Part II: Finite lamination subgroups
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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.
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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.
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.
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.
Using a probabilistic argument we show that the second bounded cohomology of an acylindrically hyperbolic group $G$ (e.g., a non-elementary hyperbolic or relatively hyperbolic group, non-exceptional mapping class group, ${rm Out}(F_n)$, dots) embeds via the natural restriction maps into the inverse limit of the second bounded cohomologies of its virtually free subgroups, and in fact even into the inverse limit of the second bounded cohomologies of its hyperbolically embedded virtually free subgroups. This result is new and non-trivial even in the case where $G$ is a (non-free) hyperbolic group. The corresponding statement fails in general for the third bounded cohomology, even for surface groups.
For any finite collection $f_i$ of fully irreducible automorphisms of the free group $F_n$ we construct a connected $delta$-hyperbolic $Out(F_n)$-complex in which each $f_i$ has positive translation length.
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