Complete outer automorphism groups of free nilpotent groups

TL;DR

The paper addresses the completeness of the outer automorphism group $\mathrm{Out}(N)$ for infinitely generated free nilpotent groups $N$ of class two. It leverages definability within $\mathrm{Out}(N)$, notably of $\mathrm{IA}(N)$ and A-symmetries, and uses extremal involutions and A-basis modelling to recover $\mathrm{Aut}(N)$ from $\mathrm{Out}(N)$. The main result is that $\mathrm{Out}(N)$ is complete, extending Baumslag-type completeness results to this infinite-rank setting. The work provides a framework based on definable subgroups and canonical involution structures to analyze symmetry in relatively free groups, with potential implications for other infinite-rank automorphism problems.

Abstract

We prove that the outer automorphism group $\mathrm{Out}(N)$ of an infinitely generated free nilpotent group $N$ of class two is complete.

Theorems & Definitions (26)

• Theorem \ref{MainTh}
• Claim 2.1
• proof
• Lemma 2.2
• proof
• Corollary 2.3
• proof
• Proposition 2.4
• proof
• Proposition 2.5