Hyperbolic Actions and 2nd Bounded Cohomology of Subgroups of 𝖮𝗎𝗍(𝖥_{𝗇})

In this two part work we prove that for every finitely generated subgroup Γ > O u t ( F n ) \Gamma >{\mathsf {Out}}(F_n) , either Γ \Gamma is virtually abelian or H b 2 ( Γ ; R ) H^2_b(\Gamma ;{\mathbb {R}}) contains a vector space embedding of ℓ 1 \ell ^1 . The method uses actions on hyperbolic spaces. In Part I we focus on the case of infinite lamination subgroups Γ \Gamma —those for which the set of all attracting laminations of all elements of Γ \Gamma is an infinite set—using actions on free splitting complexes of free groups. In Part II we focus on finite lamination subgroups Γ \Gamma and on the construction of useful new hyperbolic actions of those subgroups.