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Linearly ordered group

Group with translationally invariant total order; i.e. if a ≤ b, then ca ≤ cb From Wikipedia, the free encyclopedia

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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.

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Further definitions

In this section, is a left-invariant order on a group with identity element . All that is said applies to right-invariant orders with the obvious modifications. Note that being left-invariant is equivalent to the order defined by if and only if 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 of an ordered group positive if . The set of positive elements in an ordered group is called the positive cone, it is often denoted with ; the slightly different notation is used for the positive cone together with the identity element.[1]

The positive cone characterises the order ; indeed, by left-invariance we see that if and only if . In fact, a left-ordered group can be defined as a group together with a subset satisfying the two conditions that:

  1. for we have also ;
  2. let , then is the disjoint union of and .

The order associated with is defined by ; the first condition amounts to left-invariance and the second to the order being well-defined and total. The positive cone of is .

The left-invariant order is bi-invariant if and only if it is conjugacy-invariant, that is if then for any we have as well. This is equivalent to the positive cone being stable under inner automorphisms.


If [citation needed], then the absolute value of , denoted by , is defined to be: If in addition the group is abelian, then for any a triangle inequality is satisfied: .

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Examples

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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, of the closure of a l.o. group under th roots. We endow this space with the usual topology of a linear order, and then it can be shown that for each the exponential maps 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 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 as 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 are not left-orderable;[10] a wide generalisation of this was announced in 2020.[11]

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See also

Notes

References

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