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

Abstract

We prove that, for any positive integer $c$, the quotient group ${\gamma }_c(M_3)/{\gamma }_{c+1}(M_3)$ of the lower central series of the McCool group $M_3$ is isomorphic to two copies of the quotient group ${\gamma }_c(F_3)/{\gamma }_{c+1}(F_3)$ of the lower central series of a free group $F_3$ of rank $3$ as $\mathbb{Z}$-modules. Furthermore, we give a necessary and sufficient condition whether the associated graded Lie algebra $\mathrm{gr}(M_3)$ of $M_3$ is naturally embedded into the Johnson Lie algebra $\mathcal{L}(\mathrm{IA}(F_3))$ of the IA-automorphisms of $F_3$.

14 pages