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Groups, Orders, and Dynamics

B. Deroin, A. Navas, C. Rivas

TL;DR

This work provides a comprehensive examination of left-orderable and bi-orderable groups through a dynamical lens, treating orders as actions on ordered spaces and developing the LO(Γ) landscape, including when LO(Γ) is a Cantor set or finite. Central results link classical theorems (Hölder, Conrad) with modern dynamical techniques (dynamical realizations, crossings), and extend to probabilistic and geometric viewpoints. The text surveys concrete families (Abelian, nilpotent, free, Thompson F, braid groups) and structural tools (convex extensions, free products, Tararin theory) to produce and classify left-orders, while exploring isoperimetric connections and implications for Kaplansky-type questions. Together, these themes illuminate how orderability intersects with dynamics, topology, and algebra, with broad applications and several open problems highlighted for future work.

Abstract

This is an almost self-contained monograph (containing some new results) on left-orderable groups which mostly rely on dynamical and probabilistic aspects, but also on geometric, combinatorial, analytic, and topological ones. This new version contains many improvements, corrections and updates, many of them suggested by the referees and colleagues along the last years.

Groups, Orders, and Dynamics

TL;DR

This work provides a comprehensive examination of left-orderable and bi-orderable groups through a dynamical lens, treating orders as actions on ordered spaces and developing the LO(Γ) landscape, including when LO(Γ) is a Cantor set or finite. Central results link classical theorems (Hölder, Conrad) with modern dynamical techniques (dynamical realizations, crossings), and extend to probabilistic and geometric viewpoints. The text surveys concrete families (Abelian, nilpotent, free, Thompson F, braid groups) and structural tools (convex extensions, free products, Tararin theory) to produce and classify left-orders, while exploring isoperimetric connections and implications for Kaplansky-type questions. Together, these themes illuminate how orderability intersects with dynamics, topology, and algebra, with broad applications and several open problems highlighted for future work.

Abstract

This is an almost self-contained monograph (containing some new results) on left-orderable groups which mostly rely on dynamical and probabilistic aspects, but also on geometric, combinatorial, analytic, and topological ones. This new version contains many improvements, corrections and updates, many of them suggested by the referees and colleagues along the last years.

Paper Structure

This paper contains 68 sections, 148 theorems, 456 equations, 4 figures.

Table of Contents

  1. SOME BASIC AND NOT SO BASIC FACTS
  2. General Definitions
  3. Positive and negative cones
  4. A characterization involving finite subsets
  5. Left-orderable groups and actions on ordered spaces
  6. Semiconjugacy in $\mathrm{Homeo}_+(\mathbb R)$
  7. Some Relevant Examples
  8. Abelian and nilpotent groups
  9. Subgroups of the affine group
  10. Free and residually free groups
  11. Thompson's group $\mathrm{F}$
  12. Some relatives of F
  13. Braid groups
  14. Other Forms of Orderability
  15. Lattice-orderable groups
  16. ...and 53 more sections

Key Result

Proposition 1.1.6

Let $\Gamma$ be a group and $\{ P_{\lambda}$, $\lambda \in \Lambda\}$ a family of subsemigroups such that: (i) $P_{\lambda} \cup P_{\lambda}^{-1} = \Gamma$, for all$\lambda \in \Lambda$; (ii) The intersection$P := \bigcap_{\lambda \in \Lambda} P_{\lambda}$satisfies$P \cap P^{-1} = \{id\}$. Th

Figures (4)

  • Figure :
  • Figure :
  • Figure :
  • Figure :

Theorems & Definitions (243)

  • Example 1.1.1
  • Remark 1.1.2
  • Remark 1.1.3
  • Remark 1.1.4
  • Proposition 1.1.6
  • Proposition 1.1.10
  • Remark 1.1.11
  • Remark 1.1.12
  • Remark 1.1.14
  • Remark 1.1.15
  • ...and 233 more