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$R_{\infty}$-property for finitely generated torsion-free 2-step nilpotent groups of small Hirsch length

Karel Dekimpe, Maarten Lathouwers

Abstract

In this paper we will show that finitely generated torsion-free 2-step nilpotent groups of Hirsch length at most 6 do not have the $R_{\infty}$-property, while there are examples of such groups of Hirsch length 7 that do have the $R_{\infty}$-property.

$R_{\infty}$-property for finitely generated torsion-free 2-step nilpotent groups of small Hirsch length

Abstract

In this paper we will show that finitely generated torsion-free 2-step nilpotent groups of Hirsch length at most 6 do not have the -property, while there are examples of such groups of Hirsch length 7 that do have the -property.
Paper Structure (7 sections, 15 theorems, 51 equations)

This paper contains 7 sections, 15 theorems, 51 equations.

Table of Contents

  1. Preliminaries
  2. Hirsch length and I(n,m) groups
  3. Twisted conjugacy
  4. Reidemeister spectrum of groups in I(n,1)
  5. Hirsch length 5 or less
  6. Hirsch length 6
  7. Conclusion and minimal example

Key Result

Lemma 1.1

If $G$ is a polycyclic group (e.g. a finitely generated nilpotent group), then the Hirsch length is well-defined (i.e. independent of the chosen series). Moreover, if we fix some $H\subseteq G$ and $N\lhd G$, then the following holds:

Theorems & Definitions (27)

  • Lemma 1.1: sega83
  • Lemma 1.2: mks66
  • Definition 1.3
  • Theorem 1.4: dg14 and roma11
  • Theorem 1.5: dgo21 and roma11
  • Lemma 1.6
  • Proposition 1.7
  • proof
  • Proposition 2.1: sega83
  • Theorem 2.2
  • ...and 17 more