# Quotient groups of IA-automorphisms of a free group of rank 3

@article{Metaftsis2018QuotientGO, title={Quotient groups of IA-automorphisms of a free group of rank 3}, author={V. Metaftsis and Athanassios I. Papistas and I. Sevaslidou}, journal={Int. J. Algebra Comput.}, year={2018}, volume={28}, pages={115-131} }

We prove that, for any positive integer c, the quotient group γc(M3)/γc+1(M3) of the lower central series of the McCool group M3 is isomorphic to two copies of the quotient group γc(F3)/γc+1(F3) of the lower central series of a free group F3 of rank 3 as ℤ-modules. Furthermore, we give a necessary and sufficient condition whether the associated graded Lie algebra gr(M3) of M3 is naturally embedded into the Johnson Lie algebra ℒ(IA(F3)) of the IA-automorphisms of F3.

## 2 Citations

On certain subgroups of the McCool group

- MathematicsInt. J. Algebra Comput.
- 2020

A subgroup of the automorphism group of Fn consisting of all automorphisms which induce the identity on Fn is assigned to IA(Fn), where Fn is a free group of rank n and IA (Fn) is the Automorphism Group.

## References

SHOWING 1-10 OF 20 REFERENCES

THE COKERNEL OF THE JOHNSON HOMOMORPHISMS OF THE AUTOMORPHISM GROUP OF A FREE METABELIAN GROUP(Approach from Transformation Group Theory to Borsuk-Ulam type theorems)

- Mathematics
- 2008

In this paper, we determine the cokernel of the k-th Johnson homomorphisms of the automorphism group of a free metabelian group for k ≥ 2 and n ≥ 4. As a corollary, we obtain a lower bound on the…

On basis-conjugating automorphisms of free groups

- Mathematics
- 1986

Let X = {x 1, … xn } be a free generating set of the free group Fn and let H be the subgroup of Aut Fn consisting of those automorphisms α such that α(xi ) is conjugate to xi for each i = 1, 2 , …,…

Lie Powers of Modules for Groups of Prime Order

- Mathematics
- 2002

Let L(V) be the free Lie algebra on a finite‐dimensional vector space V over a field K, with homogeneous components Ln(V) for n ⩾ 1. If G is a group and V is a KG‐module, the action of G extends…

The Johnson homomorphism and the second cohomology of IAn

- Mathematics
- 2005

Let Fn be the free group on n generators. Define IAn to be group of automorphisms of Fn that act trivially on first homology. The Johnson homomorphism in this setting is a map from IAn to its…

Invariant Bases for Free Lie Rings

- Mathematics
- 2002

Let L denote the free Lie ring of rank 2r with free generators xi , yi (1 i r), and let τ be the automorphism of L that interchanges xi and yi for all i . We construct a Z-basis of L that consists of…

On the McCool group M3 and its associated Lie algebra

- Mathematics
- 2015

ABSTRACT We prove that the Lie algebra of the McCool group M3 is torsion free. As a result, we are able to give a presentation for the Lie algebra of M3. Furthermore, M3 is a Magnus group.

Combinatorial Group Theory

- Mathematics
- 2002

These notes were prepared for use by the participants in the Workshop on Algebra, Geometry and Topology held at the Australian National University, 22 January to 9 February, 1996. They have…

The extension of Johnson's homomorphism from the Torelli group to the mapping class group

- Mathematics
- 1993

Sur les groupes nilpotents et les anneaux de Lie

- Mathematics
- 1954

© Gauthier-Villars (Éditions scientifiques et médicales Elsevier), 1954, tous droits réservés. L’accès aux archives de la revue « Annales scientifiques de l’É.N.S. » (http://www.…