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Automorphism groups of combinatorial Hantzsche-Wendt groups

Rafał Lutowski, Andrzej Szczepański, Richard Weidmann

Abstract

Combinatorial Hantzsche-Wendt groups were introduced by W. Craig and P.A. Linnell. Every such a group $G_n$, where $n$ is a natural number, encodes the holonomy action of any $n+1$-dimensional Hantzsche-Wendt manifold. $G_2$ is the fundamental group of the classical Hantzsche-Wendt manifold -- the only one $3$-dimensional oriented flat manifold with non-cyclic holonomy group. In this article, we describe the structure of the automorphism and of the outer automorphism groups of combinatorial Hantzsche-Wendt groups.

Automorphism groups of combinatorial Hantzsche-Wendt groups

Abstract

Combinatorial Hantzsche-Wendt groups were introduced by W. Craig and P.A. Linnell. Every such a group , where is a natural number, encodes the holonomy action of any -dimensional Hantzsche-Wendt manifold. is the fundamental group of the classical Hantzsche-Wendt manifold -- the only one -dimensional oriented flat manifold with non-cyclic holonomy group. In this article, we describe the structure of the automorphism and of the outer automorphism groups of combinatorial Hantzsche-Wendt groups.
Paper Structure (7 sections, 21 theorems, 51 equations)

This paper contains 7 sections, 21 theorems, 51 equations.

Table of Contents

  1. Introduction
  2. Presentation of Aut(W)
  3. Translation endomorphisms of G
  4. Automorphisms of G
  5. Presentation of Aut(G)
  6. Semi-linear automorphisms of A
  7. Automorphisms of G. Second approach

Key Result

Lemma 3.1

Let $a=(x_1^2)^{{z_1}} \cdots (x_n^2)^{{z_n}}\in A$. Then $(x_ia)^2=(x_i^2)^{2{z_i}+1}$

Theorems & Definitions (42)

  • Definition 1.1
  • Lemma 3.1
  • proof
  • Lemma 3.2
  • proof
  • Lemma 3.3
  • proof
  • Corollary 3.4
  • Remark 3.5
  • Lemma 3.6
  • ...and 32 more