Linearly ordered group

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 $$\le'$$ defined by $$g \le' h$$ 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 $$g \not= e$$ 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: The order $$\le_P$$ associated with $$P$$ is defined by $$g \le_P h \Leftrightarrow g^{-1} h \in P$$; 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 $$P$$.
 * 1) for $$g, h \in P$$ we have also $$gh \in P$$;
 * 2) let $$P^{-1} = \{g^{-1}, g \in P\}$$, then $$G$$ is the disjoint union of $$P, P^{-1}$$ and $$\{e\}$$.

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 $$xgx^{-1} \le xhx^{-1}$$ 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: $$|a|:=\begin{cases}a, & \text{if }a \ge 0,\\ -a, & \text{otherwise}.\end{cases}$$ 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,. If we write the Archimedean l.o. group multiplicatively, this may be shown by considering the Dedekind completion, $$\widehat{G}$$ 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 $$g\in\widehat{G}$$ the exponential maps $$g^{\cdot}:(\mathbb{R},+)\to(\widehat{G},\cdot) :\lim_{i}q_{i}\in\mathbb{Q}\mapsto \lim_{i}g^{q_{i}}$$ 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.

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 $$\langle a, b | a^2ba^2b^{-1}, b^2ab^2a^{-1}\rangle$$ 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 $$\mathrm{SL}_n(\mathbb Z)$$ are not left-orderable; a wide generalisation of this has been recently announced.