Automorphisms of descending mod-p central series

Abstract

Given a free group Γ of finite rank n and a prime number p, denote by Γ • k the k th layer of the Stallings (• = S) or Zassenhaus (In this paper we prove that there is a non-centralwhich splits if and only if k = 1 and p is odd if • = Z or, k = 1 and (p, n) = (3, 2), (2, 2) if • = S. Moreover, if we denote by IA p (N • k ) the subgroup of Aut N • k formed by the automorphisms that acts trivially on N • 1 , then the restriction of this extension to IA p (N • k+1 ) give us a non-split central extension of groups 0 → Hom(N • 1