Linearly ordered group explained

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</matH> if and only if <math>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:

1. for

g, h \in P we have also gh \in P ;

1. let

P ^{-1} = \{g ^{-1} , 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 \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 .

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: 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 ^{⋅} :( R , + )\to(\widehat{G}, ⋅ ) :\lim _{i} q _{i} \in 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.^[1] Braid groups are also left-orderable.^[2]

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.^[3] There exists a 3-manifold group which is left-orderable but not bi-orderable^[4] (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

SL _{n(Z)} are not left-orderable;^[5] a wide generalisation of this has been recently announced.^[6]

See also

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

References

• Deroin . Bertrand . Navas . Andrés . Rivas . Cristóbal . 1408.5805 . Groups, orders and dynamics . math.GT . 2014 .

Notes and References

1. Duchamp . Gérard . Thibon . Jean-Yves . 1992. Simple orderings for free partially commutative groups . International Journal of Algebra and Computation . 2 . 3 . 351–355 . 10.1142/S0218196792000219 . 0772.20017 .
2. Book: Dehornoy . Patrick . Dynnikov . Ivan . Rolfsen . Dale . Wiest . Bert . 2002. Why are braids orderable? . Paris . Société Mathématique de France . xiii + 190 . 2-85629-135-X.
3. Boyer . Steven . Rolfsen . Dale. Wiest. Bert . 2005 . Orderable 3-manifold groups . 10.5802/aif.2098 . Annales de l'Institut Fourier . 55 . 1 . 243–288 . 1068.57001. free . math/0211110 .
4. Bergman . George . 1991 . Right orderable groups that are not locally indicable . Pacific Journal of Mathematics . 147 . 2 . 243–248 . 10.2140/pjm.1991.147.243 . 0677.06007. free .
5. Witte . Dave . 1994 . Arithmetic groups of higher \(\mathbb\)-rank cannot act on \(1\)-manifolds . Proceedings of the American Mathematical Society . 122 . 2 . 333–340 . 10.2307/2161021 . 2161021 . 0818.22006.
6. Deroin . Bertrand . Hurtado . Sebastian . 2008.10687 . Non left-orderability of lattices in higher rank semi-simple Lie groups . math.GT . 2020 .