Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Subgroups and diversity of left-orderable small cancellation groups

Markus Steenbock

Abstract

We arrange classical small cancellation constructions to produce left-orderable groups: we show that every finitely generated group is the quotient of a left-ordered small cancellation group by a finitely generated kernel (Rips construction). We give presentations of left-ordered hyperbolic small cancellation groups that are not locally indicable, and observe that the class of left-ordered small cancellation groups that are not locally indicable is quasi-isometrically diverse. Altogether, this shows that left-orderable small cancellation groups form a rich and diverse class.

Subgroups and diversity of left-orderable small cancellation groups

Abstract

We arrange classical small cancellation constructions to produce left-orderable groups: we show that every finitely generated group is the quotient of a left-ordered small cancellation group by a finitely generated kernel (Rips construction). We give presentations of left-ordered hyperbolic small cancellation groups that are not locally indicable, and observe that the class of left-ordered small cancellation groups that are not locally indicable is quasi-isometrically diverse. Altogether, this shows that left-orderable small cancellation groups form a rich and diverse class.
Paper Structure (15 sections, 30 theorems, 72 equations)

This paper contains 15 sections, 30 theorems, 72 equations.

Table of Contents

  1. Introduction
  2. Subgroups of left-orderable small cancellation groups
  3. Groups that are not locally indicable
  4. Quasi-isometric diversity
  5. Perfect groups
  6. Left-ordered groups
  7. Lexicographic orderings of the free group
  8. Explicit orderings on free products
  9. Combination theorems
  10. Small cancellation groups
  11. Groups that are not locally indicable
  12. Rips construction
  13. Rips construction via HNN-extensions
  14. Construction of left-orders
  15. Embedding groups that are not locally indicable

Key Result

Theorem 1.1

Let $Q$ be a finitely generated group. Then there is a short exact sequence of groups such that Moreover, if $Q$ is finitely presented, then $G$ can be chosen finitely presented.

Theorems & Definitions (59)

  • Theorem 1.1
  • Remark 1.2
  • Remark 1.3
  • Remark 1.4
  • Remark 1.5
  • Remark 1.6
  • Proposition 1.7
  • Proposition 1.8
  • Remark 1.9
  • Proposition 1.10
  • ...and 49 more