Automorphisms of descending mod-p central series

2019/03/11 by Riba, Ricard
#20D15 (Primary) #20D45 #20F14 #20J06 (Secondary) #FOS: Mathematics #Group Theory (math.GR)

paper · doi:10.48550/arxiv.1903.04396

Abstract

Given a free group Γ of finite rank n and a prime number p, denote by Γ_k^\bullet the k^th layer of the Stallings (\bullet=S) or Zassenhaus (\bullet=Z) p-central series, by N_k^\bullet the quotient Γ/Γ_k+1^\bullet and by L_k^\bullet the quotient Γ_k^\bullet /Γ_k+1^\bullet. In this paper we prove that there is a non-central extension of groups 0 \longrightarrow Hom(N^\bullet₁, L^\bullet_k+1) \longrightarrow Aut N^\bullet_k+1 \longrightarrow Aut 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(N^\bullet_k ) the subgroup of Aut N^\bullet_k formed by the automorphisms that acts trivially on N₁^\bullet, then the restriction of this extension to IA^p(N^\bullet_k+1) give us a non-split central extension of groups 0 \longrightarrow Hom(N^\bullet₁,L^\bullet_k+1) \longrightarrow IA^p(N^\bullet_k+1) \longrightarrow IA^p(N^\bullet_k ) \longrightarrow 1.

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