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 $\mathrm{\Gamma }<\text{Out}(F_n)$, either $\mathrm{\Gamma }$ is virtually abelian or $H_b^2(\mathrm{\Gamma };\mathbb{R})$ contains an embedding of ${\mathrm{\ell }}^1$. 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 $\mathrm{\Gamma }$ - those for which the set of all attracting laminations of all elements of $\mathrm{\Gamma }$ is infinite - using actions on free splitting complexes of free groups.