On Certain Questions of the Free Group Automorphisms Theory

Certain subgroups of the groups Aut(Fn) of automorphisms of a free group Fn are considered. Comparing Alexander polynomials of two poly-free groups Cb+4 and P4 we prove that these groups are not isomorphic, despite the fact that they have a lot of common properties. This answers the question of Cohen-Pakianathan-Vershinin-Wu from [8]. The questions of linearity of subgroups of Aut(Fn) are considered. As an application of the properties of poison groups in the sense of Formanek and Procesi, we show that the groups of the type Aut(G ∗ Z) for certain groups G and the subgroup of IA-automorphisms IA(Fn) ⊂ Aut(Fn) are not linear for n ≥ 3. This generalizes the recent result of Pettet that IA(Fn) are not linear for n ≥ 5.