Automorphisms of descending mod-p central series

Abstract

Given a free group $\Gamma$ of finite rank $n$ and a prime number $p,$ denote by ${\Gamma }_k^{\bullet }$ the $k^{\text{th}}$ layer of the Stallings ($\bullet =S$) or Zassenhaus ($\bullet =Z$) $p$-central series, by ${\mathcal{N}}_k^{\bullet }$ the quotient $\Gamma /{\Gamma }_{k+1}^{\bullet }$ and by ${\mathcal{L}}_k^{\bullet }$ the quotient ${\Gamma }_k^{\bullet }/{\Gamma }_{k+1}^{\bullet }.$ In this paper we prove that there is a non-central extension of groups $0⟶Hom({\mathcal{N}}_1^{\bullet },{\mathcal{L}}_{k+1}^{\bullet })⟶Aut{\mathcal{N}}_{k+1}^{\bullet }⟶Aut{\mathcal{N}}_k^{\bullet }⟶1,$ which splits if and only if $k=1$ and $p$ is odd if $\bullet =Z$ or, $k=1$ and $(p,n)=(3,2),(2,2)$ if $\bullet =S$. Moreover, if we denote by $I{A}^p({\mathcal{N}}_k^{\bullet })$ the subgroup of $Aut{\mathcal{N}}_k^{\bullet }$ formed by the automorphisms that acts trivially on ${\mathcal{N}}_1^{\bullet },$ then the restriction of this extension to $I{A}^p({\mathcal{N}}_{k+1}^{\bullet })$ give us a non-split central extension of groups $0⟶Hom({\mathcal{N}}_1^{\bullet },{\mathcal{L}}_{k+1}^{\bullet })⟶I{A}^p({\mathcal{N}}_{k+1}^{\bullet })⟶I{A}^p({\mathcal{N}}_k^{\bullet })⟶1.$

23 pages