Local indicability in ordered groups: Braids and elementary amenable groups

Abstract

Abstract. Groups which are locally indicable are also right-orderable, but not conversely. This paper considers a characterization of local indicability in right-ordered groups, the key concept being a property of right-ordered groups due to Conrad. Our methods answer a question regarding the Artin braid groups Bn which are known to be right-orderable. The subgroups Pn of pure braids enjoy an ordering which is invariant under multiplication on both sides, and it has been asked whether such an ordering of Pn could extend to a rightinvariant ordering of Bn. We answer this in the negative. We also give another proof of a recent result of Linnell that for elementary amenable groups, the concepts of right-orderability and local indicability coincide. Definitions and statement of resultsA right-ordered group is a pair (G, <), where G is a group, < is a strict total ordering of the elements of G, and right-invariance holds:If the ordering is also left-invariant,then we call (G, <) a bi-ordered group (also known in the literature as "totally ordered" or, simply, "ordered").A group G is said to be locally indicable if for every nontrivial finitely-generated subgroup H of G there is a nontrivial homomorphism H → Z onto the additive group of integers. Proposition 1.1. Locally indicable groups are right-orderable.Proof. Burns and Hale in [4] show that a group G is right-orderable if and only if every finitely generated nontrivial subgroup of G has a nontrivial quotient that is also right-orderable. See [16], Theorem 7.3.1,or [13], Theorem 3.2.1. Since Z is right-orderable, the proposition follows.The converse is not true. Bergman [2] gave an example of a finitely-generated right-orderable group which is perfect-its abelianization is trivial. Any homomorphism of such a group to Z must be trivial, so it is not locally indicable.

An Ordering for Groups of Pure Braids and Fibre-Type Hyperplane Arrangements

“…1 An important result of Dehornoy [8] (see also [13] and [29]) is that the full braid groups B n can be given a strict total ordering which is invariant under multiplication on one side (but not both). We will see that Dehornoy's ordering of B n and our ordering of P n are fundamentally incompatible; indeed any right-or left-ordering of B n is necessarily incompatible with any bi-ordering of P n , according to [27].…”

“…Recently, Rhemtulla and Rolfsen [27] have shown that no right-or left-invariant ordering of B n whatsoever can be bi-invariant when restricted to P n , or any other finite index subgroup.…”

The Class of Orderable Groups Is a Quasi-Variety with Undecidable Theory

“…In [1] (see also [2]), J. Howie shows that the classes of locally indicable groups and right orderable groups are quasivarieties and he also asserts that the class of locally indicable groups is a subclass of the class of right orderable groups. He posed the question of whether or not these two classes coincide.…”

“…Furthermore, we show that each of the quasi-varieties of orderable groups and right-orderable groups has undecidable theory. Since the paper of Howie, there has been a significnat amount of work on orderable groups (See [2] [3] [4] and the references there). We note the classic result that free groups are orderable.…”

Poly-freeness in large even Artin groups

“…every non-trivial finitely generated subgroup admits an epimorphism onto Z), and locally indicable groups are right-orderable (i.e. admit a total order which is invariant under right multiplication), see Rhemtulla-Rolfsen [15]. Some families of Artin groups are known to be poly-free.…”