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 \longrightarrow Hom(\mathcal{N}^\bullet_1, \mathcal{L}^\bullet_{k+1}) \longrightarrow Aut\;\mathcal{N}^\bullet_{k+1} \longrightarrow Aut \;\mathcal{N}^\bullet_k \longrightarrow 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 $IA^p(\mathcal{N}^\bullet_k )$ the subgroup of $Aut \;\mathcal{N}^\bullet_k$ formed by the automorphisms that acts trivially on $\mathcal{N}_1^\bullet,$ then the restriction of this extension to $IA^p(\mathcal{N}^\bullet_{k+1})$ give us a non-split central extension of groups $ 0 \longrightarrow Hom(\mathcal{N}^\bullet_1,\mathcal{L}^\bullet_{k+1}) \longrightarrow IA^p(\mathcal{N}^\bullet_{k+1}) \longrightarrow IA^p(\mathcal{N}^\bullet_k ) \longrightarrow 1. $