On groups with infinitely many right orders

Zenkov, A. V.

doi:10.1007/bf02674902

Cited by 7 publications

(9 citation statements)

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“…, k} and every i ∈ N. Both cases being similar, we will consider only the former one. Following Zenkov [66], for each i ∈ N and each ω = (ℓ 1 , . .…”

Section: The Case Of Conradian Ordersmentioning

confidence: 99%

On the dynamics of (left) orderable groups

Navas¹

2010

Ann. inst. Fourier

184

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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.Résumé. Nous développons des méthodes dynamiques pourétudier les groupes ordonnables ainsi que leurs espaces d'ordres associés. Nous donnons des preuves nouvelles etélémentaires de théorèmes dûsà Linnell (si un groupe ordonnable possède une infinité d'ordres, alors il possède une infinité non dénombrable) et McCleary (l'espace des ordres du groupe libre est un Cantor). Nous montrons que ce dernier résultat est valable aussi pour les groupes nilpotents dénombrables et sans torsion qui ne sont pas abéliens de rang un. Finalement, nous appliquons nos méthodes au cas des groupes de tresses. En particulier, nous démontrons que le cone positif de l'ordre de Dehornoy n'est pas de type fini en tant que semi-groupe. Pour ce faire, nous définissons le noyau conradien d'un ordre commeétant le plus grand sous-groupe convexe sur lequel la relation est conradienne, et nous travaillons avec cette notion.

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“…, k} and every i ∈ N. Both cases being similar, we will consider only the former one. Following Zenkov [66], for each i ∈ N and each ω = (ℓ 1 , . .…”

Section: The Case Of Conradian Ordersmentioning

confidence: 99%

On the dynamics of (left) orderable groups

Navas¹

2010

Ann. inst. Fourier

184

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“…Assuming Orb G (P ) is finite, we must have that Orb G (P ) is finite, and hence the stabilizer Stab G (P ) is a finite index subgroup of G that is biordered by the restriction of the left ordering < P . It follows that G is locally indicable [21], and hence has uncountably many left orderings, by [23].…”

Section: Now In Any Left Orderingmentioning

confidence: 99%

Free lattice-ordered groups and the space of left orderings

Clay

2011

Monatsh Math

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For any left orderable group G, we recall from work of Mc-Cleary that isolated points in the space LO(G) correspond to basic elements in the free lattice ordered group F (G). We then establish a new connection between the kernels of certain maps in the free lattice ordered group F (G), and the topology on the space of left orderings LO(G). This connection yields a simple proof that no left orderable group has countably many left orderings.When we take G to be the free group Fn of rank n, this connection sheds new light on the space of left orderings LO(Fn): by applying a result of Kopytov, we show that there exists a left ordering of the free group whose orbit is dense in the space of left orderings. From this, we obtain a new proof that LO(Fn) contains no isolated points, and equivalently, a new proof that F (Fn) contains no basic elements.

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“…This case was settled in [13, Proposition 4.1] using ideas going back to Zenkov [15] and Tararin (see Kopytov and Medvedev [8]). …”

Section: Casementioning

confidence: 99%

“…Linnell's proof uses an argument from General Topology for reducing the general case to that of Conradian orderings for which prior arguments by Zenkov [15] apply. To deal with the non Conradian case, we use our machinery on Conradian souls.…”

Section: Introductionmentioning

confidence: 99%

“…As a final application of our methods, we give a new proof of a theorem first established by Linnell [6]: if a group has infinitely many orderings, then it has uncountably many. Linnell's proof uses an argument from General Topology for reducing the general case to that of Conradian orderings for which prior arguments by Zenkov [12] apply. To deal with the non Conradian case, we use our machinery on Conradian souls.…”

Section: Introductionmentioning

confidence: 99%

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A new characterization of Conrad’s property for group orderings, with applications

Clay¹,

Navas²,

Rivas³

2009

Algebr. Geom. Topol.

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A new characterization of Conrad's property for group orderings, with applications ANDRÉS NAVAS CRISTÓBAL RIVAS APPENDIX BY ADAM CLAYWe provide a pure algebraic version of the first-named author's dynamical characterization of the Conrad property for group orderings. This approach allows dealing with general group actions on totally ordered spaces. As an application, we give a new and somehow constructive proof of a theorem first established by Linnell: an orderable group having infinitely many orderings has uncountably many. This proof is achieved by extending to uncountable orderable groups a result about orderings which may be approximated by their conjugates. This last result is illustrated by an example of an exotic ordering on the free group given by the third author in the Appendix. 06F15, 20F60; 57S25 IntroductionIn recent years, relevant progress has been made in the theory of (left) orderable groups. This has been achieved mainly by means of the use of a recently introduced mathematical object, namely the space of group orderings (see for instance Clay [2], Linnell [9], Morris [11], Navas [13] and Sikora [14]). This space may be endowed with a natural topology (roughly, two orderings are close if they coincide over large finite sets), and the study of this topological structure should reveal some algebraic features of the underlying group. In [13] it was realized that, for this study, the classical Conrad property for group orderings becomes relevant. Bringing ideas and techniques from the theory of codimension-one foliations, the 'dynamical' insight of this property was revealed. Unfortunately, many proofs of [13] are difficult to read for people with a pure algebraic view of orderable groups. More importantly, some of the results therein do not cover the case of uncountable groups. Indeed, the dynamical analysis of group orderings is done via the so-called 'dynamical realization' of orderable groups as groups of homeomorphisms of the line, which is not available for general uncountable orderable groups. Motivated by this, we develop here an algebraic counterpart of (part of) the analysis of [13]. We begin by giving a new characterization of the Conrad property that is purely algebraic, although it has a dynamical flavor (see Theorem 2.4). This leads naturally to the notion of Conradian actions on totally ordered spaces. A relevant example concerns the action of an ordered group on the space of cosets with respect to a convex subgroup.In this setting, we define the notion of Conradian extension (see Example 2.11), and we generalize Conrad's classical theorem on the 'level' structure of groups admitting Conradian orderings (see Theorem 2.14, Corollary 2.15).A relevant concept introduced in [13] is the Conradian soul, which corresponds to the maximal subgroup of an ordered group that is convex and restricted to which the ordering is Conradian. In [13], a more geometrical view of this notion was given in the case of countable groups. Here we provide an analogous algebraic description which applies to gen...

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On groups with infinitely many right orders

Cited by 7 publications

References 1 publication

On the dynamics of (left) orderable groups

On the dynamics of (left) orderable groups

Free lattice-ordered groups and the space of left orderings

A new characterization of Conrad’s property for group orderings, with applications

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