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Generalized torsion in amalgams

Tommy Wuxing Cai, Adam Clay

TL;DR

This work investigates how generalized torsion interacts with orderability in amalgams. It provides a robust sufficient condition for a free product with amalgamation to be generalized torsion-free, using a taming framework to bound products of conjugates and connect to stable commutator length techniques. Leveraging this criterion, the authors construct three explicit GTF phenomena that defy simple orderability expectations: a closed 3-manifold group that is GTF but not bi-orderable, a one-relator GTF group that is not bi-orderable, and a GTF group that is not left-orderable. These results illuminate the subtle boundaries between generalized torsion and various orderings, while offering structural tools (notably multi-malnormality and taming) applicable to broader amalgam constructions in geometric group theory.

Abstract

We give a condition sufficient to ensure that an amalgam of groups is generalized torsion-free. As applications, we construct a closed 3-manifold whose fundamental group is generalized torsion-free and non bi-orderable; a one-relator group which is generalized torsion-free and non bi-orderable; and a group which is generalized torsion-free and non left-orderable.

Generalized torsion in amalgams

TL;DR

This work investigates how generalized torsion interacts with orderability in amalgams. It provides a robust sufficient condition for a free product with amalgamation to be generalized torsion-free, using a taming framework to bound products of conjugates and connect to stable commutator length techniques. Leveraging this criterion, the authors construct three explicit GTF phenomena that defy simple orderability expectations: a closed 3-manifold group that is GTF but not bi-orderable, a one-relator GTF group that is not bi-orderable, and a GTF group that is not left-orderable. These results illuminate the subtle boundaries between generalized torsion and various orderings, while offering structural tools (notably multi-malnormality and taming) applicable to broader amalgam constructions in geometric group theory.

Abstract

We give a condition sufficient to ensure that an amalgam of groups is generalized torsion-free. As applications, we construct a closed 3-manifold whose fundamental group is generalized torsion-free and non bi-orderable; a one-relator group which is generalized torsion-free and non bi-orderable; and a group which is generalized torsion-free and non left-orderable.

Paper Structure

This paper contains 18 sections, 63 theorems, 76 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Organization of the manuscript
  3. Notation and preparatory results
  4. Some results about products in amalgams
  5. Taming products of conjugates
  6. Generalized torsion in amalgams
  7. Examples, and an analog of a theorem of Bergman
  8. Multi-malnormality and two corollaries
  9. A 3-manifold group which is GTF and not bi-orderable
  10. A one-relator group which is GTF and not bi-orderable
  11. A group which is GTF and not left-orderable
  12. The construction of a GTF and non-left-orderable group
  13. The RTF-ness of Lg in Lg
  14. Left-first products and the origin of components
  15. Multi-malnormality of Lg in Lg
  16. ...and 3 more sections

Key Result

Theorem 1.1

There exists a closed, connected $3$-manifold whose fundamental group is GTF and not bi-orderable.

Figures (1)

  • Figure 1: The figure eight knot.

Theorems & Definitions (135)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Lemma 2.1
  • proof
  • Corollary 2.2
  • proof
  • Lemma 2.3
  • proof
  • Definition 3.1
  • ...and 125 more