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The Reidemeister spectrum of low dimensional almost-crystallographic groups

Sam Tertooy

TL;DR

This work classifies the $R_ fty$-property for 4-dimensional almost-crystallographic groups and computes Reidemeister spectra for the corresponding 3D almost-crystallographic and 4D almost-Bieberbach groups. It employs a determinant criterion on holonomy-induced automorphisms, combined with quotient-by-isolator reductions and affine representations, to distinguish which 4D groups have infinite Reidemeister numbers under all automorphisms and which admit finite $R( obreak obreak obreak varphi)$. The spectra for 3D non-crystallographic cases split into two families, with the first yielding $2\mathbb{N}\cup\{\infty\}$ and the second depending on parameter parity, while the 4D Bieberbach-like families produce spectra in arithmetic progressions (multiples of 4 or 8) possibly augmented by $\infty$. Overall, the results complete the Reidemeister spectra for low-dimensional almost-crystallographic groups and dovetail with prior findings for crystallographic and Bieberbach groups, providing a comprehensive landscape of $R_ olinebreak$-infinity and finite spectra in these families.

Abstract

We determine which non-crystallographic, almost-crystallographic groups of dimension 4 have the $R_\infty$-property. We then calculate the Reidemeister spectra of the 3-dimensional almost-crystallographic groups and the 4-dimensional almost-Bieberbach groups.

The Reidemeister spectrum of low dimensional almost-crystallographic groups

TL;DR

This work classifies the -property for 4-dimensional almost-crystallographic groups and computes Reidemeister spectra for the corresponding 3D almost-crystallographic and 4D almost-Bieberbach groups. It employs a determinant criterion on holonomy-induced automorphisms, combined with quotient-by-isolator reductions and affine representations, to distinguish which 4D groups have infinite Reidemeister numbers under all automorphisms and which admit finite . The spectra for 3D non-crystallographic cases split into two families, with the first yielding and the second depending on parameter parity, while the 4D Bieberbach-like families produce spectra in arithmetic progressions (multiples of 4 or 8) possibly augmented by . Overall, the results complete the Reidemeister spectra for low-dimensional almost-crystallographic groups and dovetail with prior findings for crystallographic and Bieberbach groups, providing a comprehensive landscape of -infinity and finite spectra in these families.

Abstract

We determine which non-crystallographic, almost-crystallographic groups of dimension 4 have the -property. We then calculate the Reidemeister spectra of the 3-dimensional almost-crystallographic groups and the 4-dimensional almost-Bieberbach groups.

Paper Structure

This paper contains 20 sections, 8 theorems, 44 equations, 14 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Reidemeister numbers and spectra
  3. Almost-crystallographic groups
  4. The $R_\infty$-property for 4-dimensional al-most-crys-tal-lo-graphic groups
  5. Nilpotency class $2$
  6. Family 1.
  7. Family 2.
  8. Families 3, 4 and 5.
  9. Families 143 and 146.
  10. Nilpotency class $3$
  11. The Reidemeister spectra of the 3-dimensional al-most-crys-tal-lo-gra-phic groups
  12. Family 1.
  13. Family 2.
  14. Spectra of 4D almost-Bieberbach groups
  15. Family 1.
  16. ...and 5 more sections

Key Result

Lemma 2.1

Let $N$ be a normal subgroup of a group $G$ and $\varphi \in \mathop{\mathrm{Aut}}\nolimits(G)$ with $\varphi(N) = N$. We denote the restriction of $\varphi$ to $N$ by $\varphi|_N$, and the induced automorphism on the quotient $G/N$ by $\varphi'$. We then get the following commutative diagram with e We obtain the following properties:

Theorems & Definitions (10)

  • Lemma 2.1: see ft15-1, gw09-2
  • Corollary 2.2
  • Theorem 3.1: generalised second Bieberbach theorem
  • Definition 3.2
  • Lemma 3.3: see deki96-1
  • Theorem 3.4: see dp11-1
  • Theorem 3.5: averaging formula, see hlp12-1 and ll09-1
  • Proposition 4.1
  • proof
  • Proposition 4.2