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On the trivial units property and the unique product property

Heiko Dietrich, Melissa Lee, Andre Nies, Marc Vinyals

Abstract

We report on some computational experiments related to the trivial units property and unique product property for group rings of torsion-free groups. These properties are related to Kaplansky's unit and zero-divisor conjectures. Our investigations include a classification of certain symmetric non-trivial units in the binary group ring of the Hantzsche-Wendt group; this group was used in Gardam's refutal of Kaplansky's unit conjecture. We also exhibit and investigate a new candidate group that fails the unique units property but may satisfy the trivial unit property. No examples of groups with these properties are known to date.

On the trivial units property and the unique product property

Abstract

We report on some computational experiments related to the trivial units property and unique product property for group rings of torsion-free groups. These properties are related to Kaplansky's unit and zero-divisor conjectures. Our investigations include a classification of certain symmetric non-trivial units in the binary group ring of the Hantzsche-Wendt group; this group was used in Gardam's refutal of Kaplansky's unit conjecture. We also exhibit and investigate a new candidate group that fails the unique units property but may satisfy the trivial unit property. No examples of groups with these properties are known to date.
Paper Structure (20 sections, 8 theorems, 29 equations, 2 figures, 3 tables)

This paper contains 20 sections, 8 theorems, 29 equations, 2 figures, 3 tables.

Table of Contents

  1. Introduction
  2. Mathematical background
  3. Group rings
  4. Some background on the unit conjecture
  5. Background on the unique product property
  6. Implications between conditions
  7. Experimental results on units in $P$
  8. The group $P$
  9. Two nontrivial units
  10. Automorphisms of $P$ fixing $\{a, b, a^{-1}, b^{-1}\}$
  11. The nontrivial units of $\mathbb{F}_2[P]$ on a ball of radius 4
  12. Swap units of $\mathbb{F}_2[P]$ on balls of radius 5 and 6
  13. Fibonacci groups
  14. The groups $H_n=F(n-1,n)$ for $n \ge 4$
  15. Each $H_n$ has a solvable word problem
  16. ...and 5 more sections

Key Result

Proposition 2.1

Let $K$ be a field and let $G$ be a torsion-free group. Then $(1)\Rightarrow (2)\Rightarrow \ldots\Rightarrow(5)$, where $(1),\ldots,(5)$ are the following properties:

Figures (2)

  • Figure 1: GAP code for Example \ref{['ex1']} that demonstrates that $\mathbb{F}_2[P]$ has non-trivial units; the group $P$ is constructed following P.
  • Figure 2: GAP code to verify that $H_4$ does not satisfy the UPP.

Theorems & Definitions (23)

  • Definition 1.1
  • Conjecture 1.2
  • Remark 1.3
  • Definition 1.4
  • Proposition 2.1
  • proof
  • Example 3.1
  • Example 3.2
  • Definition 3.3
  • Lemma 4.1
  • ...and 13 more