  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.
