On the dynamics of (left) orderable groups

Andrés Navas[1]

  • [1] Univ. de Santiago de Chile Alameda 3363 Est. Central Santiago (Chile)

Annales de l’institut Fourier (2010)

  • Volume: 60, Issue: 5, page 1685-1740
  • ISSN: 0373-0956

Abstract

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We develop dynamical methods for studying left-orderable groups as well as the spaces of orderings associated to them. We give new and elementary proofs of theorems by Linnell (if a left-orderable group has infinitely many orderings, then it has uncountably many) and McCleary (the space of orderings of the free group is a Cantor set). We show that this last result also holds for countable torsion-free nilpotent groups which are not rank-one Abelian. Finally, we apply our methods to the case of braid groups. In particular, we show that the positive cone of the Dehornoy ordering is not finitely generated as a semigroup. To do this, we define the Conradian soul of an ordering as the maximal convex subgroup restricted to which the ordering is Conradian, and we elaborate on this notion.

How to cite

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Navas, Andrés. "On the dynamics of (left) orderable groups." Annales de l’institut Fourier 60.5 (2010): 1685-1740. <http://eudml.org/doc/116319>.

@article{Navas2010,
abstract = {We develop dynamical methods for studying left-orderable groups as well as the spaces of orderings associated to them. We give new and elementary proofs of theorems by Linnell (if a left-orderable group has infinitely many orderings, then it has uncountably many) and McCleary (the space of orderings of the free group is a Cantor set). We show that this last result also holds for countable torsion-free nilpotent groups which are not rank-one Abelian. Finally, we apply our methods to the case of braid groups. In particular, we show that the positive cone of the Dehornoy ordering is not finitely generated as a semigroup. To do this, we define the Conradian soul of an ordering as the maximal convex subgroup restricted to which the ordering is Conradian, and we elaborate on this notion.},
affiliation = {Univ. de Santiago de Chile Alameda 3363 Est. Central Santiago (Chile)},
author = {Navas, Andrés},
journal = {Annales de l’institut Fourier},
keywords = {Orderable groups; Conradian ordering; actions on the line; left-orderable groups; spaces of orderings; Conradian orders; braid groups; Dehornoy order; positive cones; actions on the real line},
language = {eng},
number = {5},
pages = {1685-1740},
publisher = {Association des Annales de l’institut Fourier},
title = {On the dynamics of (left) orderable groups},
url = {http://eudml.org/doc/116319},
volume = {60},
year = {2010},
}

TY - JOUR
AU - Navas, Andrés
TI - On the dynamics of (left) orderable groups
JO - Annales de l’institut Fourier
PY - 2010
PB - Association des Annales de l’institut Fourier
VL - 60
IS - 5
SP - 1685
EP - 1740
AB - We develop dynamical methods for studying left-orderable groups as well as the spaces of orderings associated to them. We give new and elementary proofs of theorems by Linnell (if a left-orderable group has infinitely many orderings, then it has uncountably many) and McCleary (the space of orderings of the free group is a Cantor set). We show that this last result also holds for countable torsion-free nilpotent groups which are not rank-one Abelian. Finally, we apply our methods to the case of braid groups. In particular, we show that the positive cone of the Dehornoy ordering is not finitely generated as a semigroup. To do this, we define the Conradian soul of an ordering as the maximal convex subgroup restricted to which the ordering is Conradian, and we elaborate on this notion.
LA - eng
KW - Orderable groups; Conradian ordering; actions on the line; left-orderable groups; spaces of orderings; Conradian orders; braid groups; Dehornoy order; positive cones; actions on the real line
UR - http://eudml.org/doc/116319
ER -

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