Hyperbolic actions and 2nd bounded cohomology of subgroups of $\text{Out}(F_n)$. Part I: Infinite lamination subgroups
Michael Handel, Lee Mosher
TL;DR
The paper proves that every finitely generated subgroup Γ of Out$(F_n)$ either is virtually abelian or has $H^2_b(Γ;\mathbb{R})$ containing an embedded $\ell^1$, by developing a WWPD-based framework and applying it to infinite lamination subgroups via actions on the free splitting complex ${\mathcal FS}(F_n)$. Part I establishes the general WWPD construction and the reduction to infinite lamination subgroups, while Part II completes the $H^2_b$-cohomology analysis (Theorem C). The methodology combines lamination ping-pong, relative train track technology, and a global WWPD property to produce free subgroups with WWPD actions and to derive unbounded quasimorphisms, feeding into Burger–Monod-type bounds. The results illuminate how hyperbolic actions and lamination dynamics govern the bounded cohomology of subgroups of Out$(F_n)$, with applications to rigidity phenomena for representations into Out$(F_n)$ and Bridson–Wade-type constraints. The framework offers a path to remove finite-generation hypotheses in later work and demonstrates the power of WWPD techniques beyond previously understood settings in Out$(F_n)$ and related groups.
Abstract
In this two part work we prove that for every finitely generated subgroup $Γ< \text{Out}(F_n)$, either $Γ$ is virtually abelian or $H^2_b(Γ;\mathbb{R})$ 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 $Γ$ - those for which the set of all attracting laminations of all elements of $Γ$ is infinite - using actions on free splitting complexes of free groups.