Hyperbolic actions and 2nd bounded cohomology of subgroups of $text{Out}(F_n)$. Part I: Infinite lamination subgroups

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.