On Certain Questions of the Free Group Automorphisms Theory

Abstract

Abstract. Certain subgroups of the groups Aut(F n ) of automorphisms of a free group F n are considered. Comparing Alexander polynomials of two poly-free groups Cb + 4 and P 4 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(F n ) 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(F n ) ⊂ Aut(F n ) are not linear for n ≥ 3. This generalizes the recent result of Pettet that IA(F n ) are not linear for n ≥ 5.

Chen ranks and resonance varieties of the upper McCool groups

“…This example shows that the single-variable Alexander polynomial of a finitely presented group G depends on a choice of presentation for the group, and thus is not an isomorphism-type invariant. Hence, the argument from [3] does not rule out the existence of an isomorphism P 4 PΣ + 4 . On the other hand, the (multi-variable) Alexander polynomial ∆ G is an isomorphism-type invariant for finitely presented groups G. Nevertheless, the groups P 4 and PΣ + 4 cannot be distinguished by means of the multi-variable Alexander polynomial.…”

“…In [3], Bardakov and Mikhailov attempted to prove that P 4 is not isomorphic to PΣ + 4 by showing that these two groups have different single-variable Alexander polynomials. To explain their approach (and why it does not work), consider a finitely presented group G, and let H = G ab /torsion be the maximal torsion-free abelian quotient of G. The group ring R = ZH is a Noetherian, commutative ring and a unique factorization domain.…”

“…For n ≤ 3, it was already known that the answer is yes. In [3], Bardakov and Mikhailov attempted to prove that P 4 is not isomorphic to PΣ + 4 by showing that the two groups have different single-variable Alexander polynomials. However, the single-variable Alexander polynomial depends on the choice of presentation for a group, and thus it cannot be used as an isomorphismtype invariant.…”

The Pure Braid Groups and Their Relatives

“…The fact that P n ∼ = wP + n for n ≥ 4 answers in the negative Problem 1 from [28, §10]. An alternate solution for n = 4 was given by Bardakov and Mikhailov in [9], but that solution relies on the claim that the single-variable Alexander polynomial of a finitely presented group G is an invariant of the group, a claim which is far from being true if b 1 (G) > 1.…”

On the McCool groupM3and its associated Lie algebra

“…al. [9] show that the graded Lie algebra of the upper triangular McCool group, gr(M + n ), is additively isomorphic to the direct sum of free Lie subalgebras The description of the Lie algebra gr(M n ), with n ≥ 2, which is stated as a problem in [3], is a non trivial problem. Since M 2 = IA(F 2 ) ∼ = F 2 , it is well known that gr(M 2 ) is a free Lie algebra of rank 2.…”