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 abc in G,
  • right-ordered group if ≤ is right-invariant, that is a ≤ b implies ac ≤ bc for all abc 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

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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:  .

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

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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

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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 has been recently announced.[11]

See also

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Notes

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  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. arXiv:math/0211110. 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. JSTOR 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

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