Template:Short description In mathematics, specifically abstract algebra, a linearly ordered or totally ordered group is a group G equipped with a total order "≤" that is translation-invariant. This may have different meanings. We say that (G, ≤) is a:
- left-ordered group if ≤ is left-invariant, that is a ≤ b implies ca ≤ cb for all a, b, c in G,
- right-ordered group if ≤ is right-invariant, that is a ≤ b implies ac ≤ bc for all a, b, c in G,
- bi-ordered group if ≤ is bi-invariant, that is it is both left- and right-invariant.
A group G is said to be left-orderable (or right-orderable, or bi-orderable) if there exists a left- (or right-, or bi-) invariant order on G. A simple necessary condition for a group to be left-orderable is to have no elements of finite order; however this is not a sufficient condition. It is equivalent for a group to be left- or right-orderable; however there exist left-orderable groups which are not bi-orderable.
Further definitionsEdit
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In this section <math>\le</math> is a left-invariant order on a group <math>G</math> with identity element <math>e</math>. All that is said applies to right-invariant orders with the obvious modifications. Note that <math>\le</math> being left-invariant is equivalent to the order <math>\le'</math> defined by <math>g \le' h</matH> if and only if <math>h^{-1} \le g^{-1}</math> 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 <math>g \not= e</math> of an ordered group positive if <math>e \le g</math>. The set of positive elements in an ordered group is called the positive cone, it is often denoted with <math>G_+</math>; the slightly different notation <math>G^+</math> is used for the positive cone together with the identity element.Template:Sfn
The positive cone <math>G_+</math> characterises the order <math>\le</math>; indeed, by left-invariance we see that <math>g \le h</math> if and only if <math>g^{-1} h \in G_+</math>. In fact a left-ordered group can be defined as a group <math>G</math> together with a subset <math>P</math> satisfying the two conditions that:
- for <math>g, h \in P</math> we have also <math>gh \in P</math>;
- let <math>P^{-1} = \{g^{-1}, g \in P\}</math>, then <math>G</math> is the disjoint union of <math>P, P^{-1}</math> and <math>\{e\}</math>.
The order <math>\le_P</math> associated with <math>P</math> is defined by <math>g \le_P h \Leftrightarrow g^{-1} h \in P</math>; the first condition amounts to left-invariance and the second to the order being well-defined and total. The positive cone of <math>\le_P</math> is <math>P</math>.
The left-invariant order <math>\le</math> is bi-invariant if and only if it is conjugacy invariant, that is if <math>g \le h</math> then for any <math>x \in G</math> we have <math>xgx^{-1} \le xhx^{-1}</math> as well. This is equivalent to the positive cone being stable under inner automorphisms.
If <math>a \in G</math>Template:Cn, then the absolute value of <math>a</math>, denoted by <math>|a|</math>, is defined to be: <math display=block>|a|:=\begin{cases}a, & \text{if }a \ge 0,\\ -a, & \text{otherwise}.\end{cases}</math>
If in addition the group <math>G</math> is abelian, then for any <math>a, b \in G</math> a triangle inequality is satisfied: <math>|a+b| \le |a|+|b|</math>.
ExamplesEdit
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;Template:Sfn this is still true for nilpotent groupsTemplate:Sfn but there exist torsion-free, finitely presented groups which are not left-orderable.
Archimedean ordered groupsEdit
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, Template:Harv. If we write the Archimedean l.o. group multiplicatively, this may be shown by considering the 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 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: which is related to the order type of the largest sequence of convex subgroups.
Other examplesEdit
Free groups are left-orderable. More generally this is also the case for right-angled Artin groups.<ref>Template:Cite journal</ref> Braid groups are also left-orderable.<ref>Template:Cite book</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;Template:Sfn 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.<ref>Template:Cite journal</ref> There exists a 3-manifold group which is left-orderable but not bi-orderable<ref> Template:Cite journal</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 systems as it is known that a countable group is left-orderable if and only if it acts on the real line by homeomorphisms.Template:Sfn Non-examples related to this paradigm are lattices in higher rank Lie groups; it is known that (for example) finite-index subgroups in <math>\mathrm{SL}_n(\mathbb Z)</math> are not left-orderable;<ref>Template:Cite journal</ref> a wide generalisation of this has been recently announced.<ref>Template:Cite arXiv</ref>
See alsoEdit
NotesEdit
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