Separating subgroups of Aut(F_n) using homological representations

Abstract

Consider the free group on n generators, denoted Fn. Then Aut(Fn) is not a lineargroup, but one can construct many finite dimensional representations using finite index characteristic subgroups of Fn. We use these types of representations to distinguish consecutive terms in the Johnson filtration of Aut(Fn). This paper relates the action described from the Johnson filtration to the homology of some subgroup K ≤ Fn. These techniques can be used to separate specific types of subgroups B ≤ A ≤ Aut(Fn).