linearly ordered group

{{more footnotes needed|section|date=July 2024}}

In this section $\backslash le$ is a left-invariant order on a group $G$ with identity element $e$. All that is said applies to right-invariant orders with the obvious modifications. Note that $\backslash le$ being left-invariant is equivalent to the order $\backslash le\text{'}$ defined by $g\; \backslash le\text{'}\; h$ if and only if $h^\{-1\}\; \backslash le\; g^\{-1\}$ being right-invariant. In particular a group being left-orderable is the same as it being right-orderable.

In analogy with ordinary numbers we call an element $g\; \backslash not=\; e$ of an ordered group positive if $e\; \backslash le\; g$. The set of positive elements in an ordered group is called the positive cone, it is often denoted with $G\_+$; the slightly different notation $G^+$ is used for the positive cone together with the identity element.{{sfn|Deroin|Navas|Rivas|2014|loc=1.1.1}}

The positive cone $G\_+$ characterises the order $\backslash le$; indeed, by left-invariance we see that $g\; \backslash le\; h$ if and only if $g^\{-1\}\; h\; \backslash in\; G\_+$. In fact a left-ordered group can be defined as a group $G$ together with a subset $P$ satisfying the two conditions that:

1. for $g,\; h\; \backslash in\; P$ we have also $gh\; \backslash in\; P$;
2. let $P^\{-1\}\; =\; \backslash \{g^\{-1\},\; g\; \backslash in\; P\backslash \}$, then $G$ is the disjoint union of $P,\; P^\{-1\}$ and $\backslash \{e\backslash \}$.

The order $\backslash le\_P$ associated with $P$ is defined by $g\; \backslash le\_P\; h\; \backslash Leftrightarrow\; g^\{-1\}\; h\; \backslash in\; P$; the first condition amounts to left-invariance and the second to the order being well-defined and total. The positive cone of $\backslash le\_P$ is $P$.

The left-invariant order $\backslash le$ is bi-invariant if and only if it is conjugacy invariant, that is if $g\; \backslash le\; h$ then for any $x\; \backslash in\; G$ we have $xgx^\{-1\}\; \backslash le\; xhx^\{-1\}$ as well. This is equivalent to the positive cone being stable under inner automorphisms.

If $a\; \backslash in\; G${{cn|date=July 2024}}, then the absolute value of $a$, denoted by $|a|$, is defined to be:

If in addition the group $G$ is abelian, then for any $a,\; b\; \backslash in\; G$ a triangle inequality is satisfied: $|a+b|\; \backslash le\; |a|+|b|$.

Any left- or right-orderable group is 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 groups{{sfn|Deroin|Navas|Rivas|2014|loc=1.2.1}} but there exist torsion-free, finitely presented groups which are not left-orderable.

Otto Hölder showed that every Archimedean group (a bi-ordered group satisfying an Archimedean property) is isomorphic to a subgroup of the additive group of real numbers, {{harv|Fuchs|Salce|2001|p=61}}.

If we write the Archimedean l.o. group multiplicatively, this may be shown by considering the Dedekind completion, $\backslash widehat\{G\}$ of the closure of a l.o. group under $n$th roots. We endow this space with the usual topology of a linear order, and then it can be shown that for each $g\backslash in\backslash widehat\{G\}$ the exponential maps $g^\{\backslash cdot\}:(\backslash mathbb\{R\},+)\backslash to(\backslash widehat\{G\},\backslash cdot)\; :\backslash lim\_\{i\}q\_\{i\}\backslash in\backslash mathbb\{Q\}\backslash mapsto\; \backslash lim\_\{i\}g^\{q\_\{i\}\}$ 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: which is related to the order type of the largest sequence of convex subgroups.

Free groups are left-orderable. More generally this is also the case for right-angled Artin groups.{{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=}} Braid groups are also left-orderable.{{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}}

The group given by the presentation $\backslash langle\; a,\; b\; |\; a^2ba^2b^\{-1\},\; b^2ab^2a^\{-1\}\backslash rangle$ 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 unit conjecture. More generally the topic of orderability of 3--manifold groups is interesting for its relation with various topological invariants.{{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 }} There exists a 3-manifold group which is left-orderable but not bi-orderable

{{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 }} (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 systems 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 lattices in higher rank Lie groups; it is known that (for example) finite-index subgroups in $\backslash mathrm\{SL\}\_n(\backslash mathbb\; Z)$ are not left-orderable;{{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}} a wide generalisation of this has been recently announced.{{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 }}

• Cyclically ordered group
• Hahn embedding theorem
• Partially ordered group

{{reflist|group=lower-alpha}}

{{reflist}}

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• {{Citation | last1=Levi | first1=F.W. | title=Ordered groups. | journal=Proc. Indian Acad. Sci. | year=1942 | volume=A16 | issue=4 | pages=256–263| doi=10.1007/BF03174799 | s2cid=198139979 }}
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• {{Citation | last1=Ghys | first1=É. | title=Groups acting on the circle. | journal=L'Enseignement Mathématique | year=2001 | volume=47 | pages=329–407}}

Category:Ordered groups