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 ${\displaystyle G}$  with identity element ${\displaystyle e}$ . 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 ${\displaystyle g\leq 'h}$  if and only if ${\displaystyle h^{-1}\leq g^{-1}}$  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 ${\displaystyle g\not =e}$  of an ordered group positive if ${\displaystyle e\leq g}$ . The set of positive elements in an ordered group is called the positive cone, it is often denoted with ${\displaystyle G_{+}}$ ; the slightly different notation ${\displaystyle G^{+}}$  is used for the positive cone together with the identity element.^[1]

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

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

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

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

If ${\displaystyle a\in G}$ , then the absolute value of ${\displaystyle a}$ , denoted by ${\displaystyle |a|}$ , is defined to be:

$${\displaystyle |a|:={\begin{cases}a,&{\text{if }}a\geq 0,\\-a,&{\text{otherwise}}.\end{cases}}}$$

 

If in addition the group ${\displaystyle G}$  is abelian, then for any ${\displaystyle a,b\in G}$  a triangle inequality is satisfied: ${\displaystyle |a+b|\leq |a|+|b|}$ .

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;^[2] this is still true for nilpotent groups^[3] 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, ${\displaystyle {\widehat {G}}}$  of the closure of a l.o. group under ${\displaystyle n}$ th roots. We endow this space with the usual topology of a linear order, and then it can be shown that for each ${\displaystyle g\in {\widehat {G}}}$  the exponential maps ${\displaystyle g^{\cdot }:(\mathbb {R} ,+)\to ({\widehat {G}},\cdot ):\lim _{i}q_{i}\in \mathbb {Q} \mapsto \lim _{i}g^{q_{i}}}$  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.^[4] Braid groups are also left-orderable.^[5]

The group given by the presentation ${\displaystyle \langle a,b|a^{2}ba^{2}b^{-1},b^{2}ab^{2}a^{-1}\rangle }$  is torsion-free but not left-orderable;^[6] 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.^[7] There exists a 3-manifold group which is left-orderable but not bi-orderable^[8] (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 cas it is known that a countable group is left-orderable if and only if it acts on the real line by homeomorphisms.^[9] Non-examples related to this paradigm are lattices in higher rank Lie groups; it is known that (for example) finite-index subgroups in ${\displaystyle \mathrm {SL} _{n}(\mathbb {Z} )}$  are not left-orderable;^[10] a wide generalisation of this has been recently announced.^[11]

See also

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

Notes

1. Deroin, Navas & Rivas 2014, 1.1.1.
2. Levi 1942.
3. Deroin, Navas & Rivas 2014, 1.2.1.
4. Duchamp, Gérard; Thibon, Jean-Yves (1992). "Simple orderings for free partially commutative groups". International Journal of Algebra and Computation. 2 (3): 351–355. doi:10.1142/S0218196792000219. Zbl 0772.20017.
5. Dehornoy, Patrick; Dynnikov, Ivan; Rolfsen, Dale; Wiest, Bert (2002). Why are braids orderable?. Paris: Société Mathématique de France. p. xiii + 190. ISBN 2-85629-135-X.
6. Deroin, Navas & Rivas 2014, 1.4.1.
7. Boyer, Steven; Rolfsen, Dale; Wiest, Bert (2005). "Orderable 3-manifold groups". Annales de l'Institut Fourier. 55 (1): 243–288. doi:10.5802/aif.2098. Zbl 1068.57001.
8. Bergman, George (1991). "Right orderable groups that are not locally indicable". Pacific Journal of Mathematics. 147 (2): 243–248. doi:10.2140/pjm.1991.147.243. Zbl 0677.06007.
9. Deroin, Navas & Rivas 2014, Proposition 1.1.8.
10. Witte, Dave (1994). "Arithmetic groups of higher \(\mathbb{Q}\)-rank cannot act on \(1\)-manifolds". Proceedings of the American Mathematical Society. 122 (2): 333–340. doi:10.2307/2161021. Zbl 0818.22006.
11. Deroin, Bertrand; Hurtado, Sebastian (2020). "Non left-orderability of lattices in higher rank semi-simple Lie groups". arXiv:2008.10687 [math.GT].

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, vol. 84, Providence, R.I.: American Mathematical Society, ISBN 978-0-8218-1963-0, MR 1794715
• Ghys, É. (2001), "Groups acting on the circle.", L'Enseignement Mathématique, 47: 329–407