Group with translationally invariant total order

Linearly ordered group

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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, $\leq$ 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 $\leq$ being left-invariant is equivalent to the order $\leq '$ defined by $g\leq 'h$ if and only if $h^{-1}\leq g^{-1}$ 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 $g\not =e$ of an ordered group positive if $e\leq g$ . 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 $\leq$ ; indeed, by left-invariance we see that $g\leq h$ 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:

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

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

The left-invariant order $\leq$ is bi-invariant if and only if it is conjugacy-invariant, that is if $g\leq h$ then for any $x\in G$ we have $xgx^{-1}\leq xhx^{-1}$ 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: $|a|:={\begin{cases}a,&{\text{if }}a\geq 0,\\a^{-1},&{\text{otherwise}}.\end{cases}}$ If in addition the group $G$ is abelian, then for any $a,b\in G$ a triangle inequality is satisfied: $|a+b|\leq |a|+|b|$ .

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, ${\widehat {G}}$ 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 $\langle a,b|a^{2}ba^{2}b^{-1},b^{2}ab^{2}a^{-1}\rangle$ 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 $\mathrm {SL} _{n}(\mathbb {Z})$ are not left-orderable; a wide generalisation of this was announced in 2020.

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