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, ≤ {\displaystyle \leq } is a left-invariant order on a group G {\displaystyle G} with identity element e {\displaystyle e}. All that is said applies to right-invariant orders with the obvious modifications. Note that ≤ {\displaystyle \leq } being left-invariant is equivalent to the order ≤ ′ {\displaystyle \leq '} defined by g ≤ ′ h {\displaystyle g\leq 'h} if and only if h − 1 ≤ g − 1 {\displaystyle h^{-1}\leq 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 {\displaystyle g\not =e} of an ordered group positive if e ≤ g {\displaystyle e\leq g}. The set of positive elements in an ordered group is called the positive cone, it is often denoted with G + {\displaystyle G_{+}}; the slightly different notation G + {\displaystyle G^{+}} is used for the positive cone together with the identity element.

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

for g , h ∈ P {\displaystyle g,h\in P} we have also g h ∈ P {\displaystyle gh\in P};
let P − 1 = { g − 1 ∣ g ∈ P } {\displaystyle P^{-1}=\{g^{-1}\mid g\in P\}}, then G {\displaystyle G} is the disjoint union of P , P − 1 {\displaystyle P,P^{-1}} and { e } {\displaystyle \{e\}}.

The order ≤ P {\displaystyle \leq _{P}} associated with P {\displaystyle P} is defined by g ≤ P h ⇔ g − 1 h ∈ P {\displaystyle g\leq _{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 ≤ P {\displaystyle \leq _{P}} is P {\displaystyle P}.

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

If a ∈ G {\displaystyle a\in G}, then the absolute value of a {\displaystyle a}, denoted by | a | {\displaystyle |a|}, is defined to be: | a | := { a , if a ≥ 0 , a − 1 , otherwise . {\displaystyle |a|:={\begin{cases}a,&{\text{if }}a\geq 0,\\a^{-1},&{\text{otherwise}}.\end{cases}}} If in addition the group G {\displaystyle G} is abelian, then for any a , b ∈ G {\displaystyle a,b\in G} a triangle inequality is satisfied: | a + b | ≤ | a | + | b | {\displaystyle |a+b|\leq |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 2001, p. 61). If we write the Archimedean l.o. group multiplicatively, this may be shown by considering the Dedekind completion, G ^ {\displaystyle {\widehat {G}}} of the closure of a l.o. group under n {\displaystyle 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 ∈ G ^ {\displaystyle g\in {\widehat {G}}} the exponential maps g ⋅ : ( R , + ) → ( G ^ , ⋅ ) : lim i q i ∈ Q ↦ lim i g q i {\displaystyle 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 ⟨ a , b | a 2 b a 2 b − 1 , b 2 a b 2 a − 1 ⟩ {\displaystyle \langle a,b|a^{2}ba^{2}b^{-1},b^{2}ab^{2}a^{-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 S L n ( Z ) {\displaystyle \mathrm {SL} _{n}(\mathbb {Z} )} are not left-orderable; a wide generalisation of this was announced in 2020.

Notes