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 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:
- for we have also ;
- 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 , then the absolute value of , denoted by , is defined to be:
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, 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 has been recently announced.[11]
See also
Notes
- ^ Levi 1942.
- ^ 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.
- ^ 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.
- ^ 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.
- ^ 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.
- ^ 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.
- ^ 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, S2CID 198139979
- 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
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