In this two part work we prove that for every finitely generated subgroup ? < Out(Fn), either ? is virtually abelian or H2 b (?; R) contains a vector space embedding of 1. The method uses actions on hyperbolic spaces. In Part I we focus on the case of infinite lamination subgroups ?—those for which the set of all attracting laminations of all elements of ? is an infinite set—using actions on free splitting complexes of free groups. In Part II we focus on finite lamination subgroups ? and on the construction of useful new hyperbolic actions of those subgroups.

In this two part work we prove that for every finitely generated subgroup ? < Out(Fn), either ? is virtually abelian or H2 b (?; R) contains a vector space embedding of 1. The method uses actions on hyperbolic spaces. In Part I we focus on the case of infinite lamination subgroups ?—those for which the set of all attracting laminations of all elements of ? is an infinite set—using actions on free splitting complexes of free groups. In Part II we focus on finite lamination subgroups ? and on the construction of useful new hyperbolic actions of those subgroups.