Linearly ordered group

Short description: Group with translationally invariant total order; i.e. if a ≤ b, then ca ≤ cb

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 definitions

In this section [math]\displaystyle{ \le }[/math] is a left-invariant order on a group [math]\displaystyle{ G }[/math] with identity element [math]\displaystyle{ e }[/math]. All that is said applies to right-invariant orders with the obvious modifications. Note that [math]\displaystyle{ \le }[/math] being left-invariant is equivalent to the order [math]\displaystyle{ \le' }[/math] defined by [math]\displaystyle{ g \le' h }[/math] if and only if [math]\displaystyle{ 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]\displaystyle{ g \not= e }[/math] of an ordered group positive if [math]\displaystyle{ e \le g }[/math]. The set of positive elements in an ordered group is called the positive cone, it is often denoted with [math]\displaystyle{ G_+ }[/math]; the slightly different notation [math]\displaystyle{ G^+ }[/math] is used for the positive cone together with the identity element.

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

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

The order [math]\displaystyle{ \le_P }[/math] associated with [math]\displaystyle{ P }[/math] is defined by [math]\displaystyle{ 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]\displaystyle{ \le_P }[/math] is [math]\displaystyle{ P }[/math].

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

If [math]\displaystyle{ a \in G }[/math], then the absolute value of [math]\displaystyle{ a }[/math], denoted by [math]\displaystyle{ |a| }[/math], is defined to be: [math]\displaystyle{ |a|:=\begin{cases}a, & \text{if }a \ge 0,\\ -a, & \text{otherwise}.\end{cases} }[/math] If in addition the group [math]\displaystyle{ G }[/math] is abelian, then for any [math]\displaystyle{ a, b \in G }[/math] a triangle inequality is satisfied: [math]\displaystyle{ |a+b| \le |a|+|b| }[/math].

Examples

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; this is still true for nilpotent groups but there exist torsion-free, finitely presented groups which are not left-orderable.

Archimedean ordered groups

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, (Fuchs Salce). If we write the Archimedean l.o. group multiplicatively, this may be shown by considering the Dedekind completion, [math]\displaystyle{ \widehat{G} }[/math] of the closure of a l.o. group under [math]\displaystyle{ 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]\displaystyle{ g\in\widehat{G} }[/math] the exponential maps [math]\displaystyle{ 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 examples

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

The group given by the presentation [math]\displaystyle{ \langle a, b | a^2ba^2b^{-1}, b^2ab^2a^{-1}\rangle }[/math] is torsion-free but not left-orderable; 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. There exists a 3-manifold group which is left-orderable but not bi-orderable (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. Non-examples related to this paradigm are lattices in higher rank Lie groups; it is known that (for example) finite-index subgroups in [math]\displaystyle{ \mathrm{SL}_n(\mathbb Z) }[/math] are not left-orderable; a wide generalisation of this has been recently announced.

See also

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

References

• Deroin, Bertrand; Navas, Andrés; Rivas, Cristóbal (2014). "Groups, orders and dynamics". arXiv:1408.5805 [math.GT].
• Levi, F.W. (1942), "Ordered groups.", Proc. Indian Acad. Sci. A16 (4): 256–263, doi:10.1007/BF03174799 
• Fuchs, László; Salce, Luigi (2001), Modules over non-Noetherian domains, Mathematical Surveys and Monographs, 84, Providence, R.I.: American Mathematical Society, ISBN 978-0-8218-1963-0 
• Ghys, É. (2001), "Groups acting on the circle.", L'Enseignement Mathématique 47: 329–407