Editing Linearly ordered group (section)

==Examples==

Any left- or right-orderable group is [[Torsion (algebra)|torsion]]-free, that is it contains no elements of finite order besides the identity. Conversely, [[F. W. Levi]] showed that a torsion-free [[abelian group]] is bi-orderable;{{sfn|Levi|1942}} this is still true for [[nilpotent group]]s{{sfn|Deroin|Navas|Rivas|2014|loc=1.2.1}} but there exist torsion-free, [[finitely presented group]]s which are not left-orderable. 

===Archimedean ordered groups===

[[Otto Hölder]] showed that every [[Archimedean group]] (a bi-ordered group satisfying an [[Archimedean property]]) is [[isomorphism|isomorphic]] to a [[subgroup]] of the additive group of [[real number]]s, {{harv|Fuchs|Salce|2001|p=61}}.
If we write the Archimedean l.o. group multiplicatively, this may be shown by considering the [[Dedekind–MacNeille completion|Dedekind completion]], <math>\widehat{G}</math> of the closure of a l.o. group under <math>n</math>th roots. We endow this space with the usual [[Topological_space#Definitions|topology]] of a linear order, and then it can be shown that for each <math>g\in\widehat{G}</math> the exponential maps <math>g^{\cdot}:(\mathbb{R},+)\to(\widehat{G},\cdot) :\lim_{i}q_{i}\in\mathbb{Q}\mapsto \lim_{i}g^{q_{i}}</math> are well defined order preserving/reversing, [[topological group]] isomorphisms. Completing a l.o. group can be difficult in the non-Archimedean case. In these cases, one may classify a group by its [[Rank of a group|rank]]: which is related to the order type of the largest sequence of convex subgroups.

===Other examples===

[[Free group]]s are left-orderable. More generally this is also the case for [[right-angled Artin group]]s.<ref>{{cite journal |last1=Duchamp |first1=Gérard |last2=Thibon |first2=Jean-Yves |date= 1992|title=Simple orderings for free partially commutative groups |url= |journal=International Journal of Algebra and Computation |volume=2 |issue=3 |pages=351–355 |doi=10.1142/S0218196792000219 |zbl=0772.20017 |access-date=}}</ref> [[Braid group]]s are also left-orderable.<ref>{{cite book |last1=Dehornoy |first1=Patrick |last2=Dynnikov |first2=Ivan |last3=Rolfsen |first3=Dale |last4=Wiest |first4=Bert | author-link= |date= 2002|title=Why are braids orderable? |url= |location=Paris |publisher=Société Mathématique de France |page=xiii + 190 |isbn=2-85629-135-X}}</ref>

The group given by the presentation <math>\langle a, b | a^2ba^2b^{-1}, b^2ab^2a^{-1}\rangle</math> is torsion-free but not left-orderable;{{sfn|Deroin|Navas|Rivas|2014|loc=1.4.1}} note that it is a 3-dimensional [[crystallographic group]] (it can be realised as the group generated by two glided half-turns with orthogonal axes and the same translation length), and it is the same group that was proven to be a counterexample to the [[Kaplansky conjectures|unit conjecture]]. More generally the topic of orderability of 3--manifold groups is interesting for its relation with various topological invariants.<ref>{{cite journal |last1=Boyer |first1=Steven |last2=Rolfsen |first2=Dale| last3=Wiest| first3=Bert |date=2005 |title=Orderable 3-manifold groups |doi=10.5802/aif.2098 |journal=Annales de l'Institut Fourier |volume=55 |issue=1 |pages=243–288 | zbl=1068.57001|doi-access=free |arxiv=math/0211110 }}</ref> There exists a 3-manifold group which is left-orderable but not bi-orderable<ref>
{{cite journal |last=Bergman |first=George |date=1991 |title=Right orderable groups that are not locally indicable |url= |journal=Pacific Journal of Mathematics |volume=147 |issue=2 |pages=243–248 |doi=10.2140/pjm.1991.147.243 |zbl=0677.06007|doi-access=free }}</ref> (in fact it does not satisfy the weaker property of being locally indicable). 

Left-orderable groups have also attracted interest from the perspective of [[dynamical system]]s as it is known that a countable group is left-orderable if and only if it acts on the real line by homeomorphisms.{{sfn|Deroin|Navas|Rivas|2014|loc=Proposition 1.1.8}} Non-examples related to this paradigm are [[Lattice (discrete subgroup)|lattices]] in higher rank [[Lie group]]s; it is known that (for example) finite-index subgroups in <math>\mathrm{SL}_n(\mathbb Z)</math> are not left-orderable;<ref>{{cite journal |last=Witte |first=Dave |date=1994 |title=Arithmetic groups of higher \(\mathbb{Q}\)-rank cannot act on \(1\)-manifolds |journal=Proceedings of the American Mathematical Society |volume=122 |issue=2 |pages=333–340 |doi=10.2307/2161021 |jstor=2161021 | zbl=0818.22006}}</ref> a wide generalisation of this has been recently announced.<ref>{{cite arXiv |last1=Deroin |first1=Bertrand |last2=Hurtado |first2=Sebastian |eprint=2008.10687 |title=Non left-orderability of lattices in higher rank semi-simple Lie groups |class=math.GT |date=2020 }}</ref>