Automorphisms of a Free Centre-by-Centre-by-Metabelian Group of Rank 3

TL;DR

The paper investigates the automorphism group of a specific free centre-by-centre-by-metabelian group of rank 3, revealing the existence of a dense, proper finitely generated subgroup within it.

Contribution

It demonstrates that the automorphism group of this complex group contains a dense, proper finitely generated subgroup, highlighting new structural properties.

Findings

Existence of a dense, proper finitely generated subgroup in Aut(G_3)

Aut(G_3) has rich subgroup structure

Provides insights into automorphisms of complex free groups

Abstract

Let $F_{3}$ be the free group of rank $3$ and let $G_{3} = F_{3}/[F_{3}^{\prime\prime}, F_{3}, F_{3}]$, that is, $G_{3}$ is a free centre-by-centre-by-metabelian group of rank $3$. We show that ${\rm Aut}(G_{3})$ contains a proper finitely generated subgroup that is dense with respect to the formal power series topology.