Linearly ordered group

Short description: Group with translationally invariant total order

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, $\le$ 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 $\le$ being left-invariant is equivalent to the order ${\le }^{\prime }$ defined by $g{\le }^{\prime }h$ if and only if $h^{-1}\le g^{-1}$ 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 $g=e$ of an ordered group positive if $e\le g$. 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.

The positive cone $G_{+}$ characterises the order $\le$; indeed, by left-invariance we see that $g\le h$ if and only if $g^{-1}h\in G_{+}$. 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 $g,h\in P$ we have also $gh\in P$;
2. let $P^{-1}=\{g^{-1}\mid g\in P\}$, then $G$ is the disjoint union of $P,P^{-1}$ and $\{e\}$.

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

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

If $a\in G$, then the absolute value of $a$, denoted by $|a|$, is defined to be: $$|a|:=\{\begin{array}{ll}a,& \text{if }a\ge 0,\\ a^{-1},& \text{otherwise}.\end{array}$$ If in addition the group $G$ is abelian, then for any $a,b\in G$ a triangle inequality is satisfied: $|a+b|\le |a|+|b|$.

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, $\hat{G}$ 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 $g\in \hat{G}$ the exponential maps $g^{\cdot }:(\mathbb{ℝ},+)\to (\hat{G},\cdot ):\underset{i}{\lim }{q}_i\in \mathbb{\mathbb{Q}}\mapsto \underset{i}{\lim }{g}^{q_i}$ 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.

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 $⟨a,b|a^{2}b{a}^2{b}^{-1},b^{2}a{b}^2{a}^{-1}⟩$ 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 ${\mathrm{S}\mathrm{L}}_n(\mathbb{\mathbb{Z}})$ are not left-orderable; a wide generalisation of this was announced in 2020.

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