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

C. E. Kofinas

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.

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

Abstract

Let be the free group of rank and let , that is, is a free centre-by-centre-by-metabelian group of rank . We show that contains a proper finitely generated subgroup that is dense with respect to the formal power series topology.

Paper Structure

This paper contains 14 sections, 17 theorems, 52 equations.

Table of Contents

  1. Introduction
  2. Preliminaries and Background
  3. Notation
  4. Automorphisms of Relatively Free Groups
  5. The Formal Power Series Topology
  6. A Dense Subgroup of ${\rm Aut}(G_{3})$
  7. Structure of the Kernel $H_{3}$
  8. Identical Relations in $G_{3}$
  9. The Kernel $H_{3}$
  10. Generating Sets of the Kernel $H_{3}$
  11. Technical Commutator Identities
  12. Further analysis of $H_{3}$
  13. A Reduction of the Generating Set of $H_{3}$
  14. Proof of Theorem \ref{['the']}

Key Result

Theorem 1

Let $G_{3} = F_{3}/[F_{3}^{\prime\prime}, F_{3}, F_{3}]$ be a free centre-by-centre-by-metabelian group of rank $3$ freely generated by the set $\{x_{1}, x_{2}, x_{3}\}$. Let ${\rm Aut}(G_{3})$ be the automorphism group of $G_{3}$ and let $T_{3}$ be the group of tame automorphisms of $G_{3}$. Then, is a proper subgroup of ${\rm Aut}(G_{3})$ that is dense in ${\rm Aut}(G_{3})$ with respect to the

Theorems & Definitions (28)

  • Theorem 1
  • Proposition 1
  • Lemma 1
  • Proposition 2
  • Proposition 3
  • Proposition 4
  • proof
  • Lemma 2
  • Lemma 3
  • Lemma 4
  • ...and 18 more