We introduce a Dehornoy-like ordering of groups, which is a generalization of the Dehornoy ordering of the braid groups. Under a weak assumption which we call Property F, we show that Dehornoy-like orderings have properties similar to the Dehornoy ordering, and produce isolated left orderings. We also construct new examples of Dehornoy-like ordering and isolated orderings and study their more precise properties.
We show that the restriction of the Dehornoy ordering to an appropriate free subgroup of the three-strand braid group defines a left ordering of the free group on k generators, k>1, that has no convex subgroups.
We introduce a condition on Garside groups that we call Dehornoy structure. An iteration of such a structure leads to a left order on the group. We show conditions for a Garside group to admit a Dehornoy structure, and we apply these criteria to prove that the Artin groups of type A and I 2 (m), m $\ge$ 4, have Dehornoy structures. We show that the left orders on the Artin groups of type A obtained from their Dehornoy structures are the Dehornoy orders. In the case of the Artin groups of type I 2 (m), m $\ge$ 4, we show that the left orders derived from their Dehornoy structures coincide with …
Dehornoy showed that the Artin braid groups $B_n$ are left-orderable. This ordering is discrete, but we show that, for $n >2$ the Dehornoy ordering, when restricted to certain natural subgroups, becomes a dense ordering. Among subgroups which arise are the commutator subgroup and the kernel of the Burau representation (for those $n$ for which the kernel is nontrivial). These results follow from a characterization of least positive elements of any normal subgroup of $B_n$ which is discretely ordered by the Dehornoy ordering.
This article surveys many standard results about the braid group with emphasis on simplifying the usual algebraic proofs. We use van der Waerden's trick to illuminate the Artin-Magnus proof of the classic presentation of the algebraic mapping-class group of a punctured disc. We give a simple, new proof of the Dehornoy-Larue braid-group trichotomy, and, hence, recover the Dehornoy right-ordering of the braid group. We then turn to the Birman-Hilden theorem concerning braid-group actions on free products of cyclic groups, and the consequences derived by Perron-Vannier, and the connections with t…
We describe the restriction of the Dehornoy ordering of braids to the dual braid monoids introduced by Birman, Ko and Lee: we give an inductive characterization of the ordering of the dual braid monoids and compute the corresponding ordinal type. The proof consists in introducing a new ordering on the dual braid monoid using the rotating normal form of arXiv:0811.3902 [math.GR], and then proving that this new ordering coincides with the standard ordering of braids.
We prove that for any finite Thurston-type ordering $<_{T}$ on the braid group\ $B_{n}$, the restriction to the positive braid monoid $(B_{n}^{+},<_{T})$ is a\ well-ordered set of order type $ω^{ω^{n-2}}$. The proof uses a combi\ natorial description of the ordering $<_{T}$. Our combinatorial description is \ based on a new normal form for positive braids which we call the $\C$-normal fo\ rm. It can be seen as a generalization of Burckel's normal form and Dehornoy's \ $Φ$-normal form (alternating normal form).
Let G be a left orderable group and LO(G) the space of all left orderings. We investigate the circumstances under which a left ordering < of G can correspond to an isolated point in LO(G), in particular we extend known results to cover the case of uncountable groups. With minor technical restrictions on the group G, we also find that no dense left ordering is isolated in LO(G), and that the closure of the set of all dense left orderings of G yields a dense G-delta set within a Cantor set of left orderings in LO(G). Lastly, we show that certain conditions on a discrete left ordering of G can gu…
We show that an amalgamated free product $G*_{A}H$ admits a discrete isolated ordering, under some assumptions of $G,H$ and $A$. This generalizes the author's previous construction of isolated orderings, and unlike known constructions of isolated orderings, can produce an isolated ordering with many non-trivial proper convex subgroups.
We give a new method to construct isolated left orderings of groups whose positive cones are finitely generated. Our construction uses an amalgamated free product of two groups having an isolated ordering. We construct a lot of new examples of isolated orderings, and give an example of isolated left orderings having various properties which previously known isolated orderings do not have.
This is a draft of a book submitted for publication by the AMS. Its theme is the remarkable interplay, accelerating in the last few decades, between topology and the theory of orderable groups, with applications in both directions. It begins with an introduction to orderable groups and their algebraic properties. Many of the algebraic results are proved by topological methods, via consideration of the space of orderings. After a discussion Hölder's theorem and some dynamical aspects of orderable groups, we provide explicit orderings of important groups in topology: free groups and most surface…
We study the space of left-orderings on groups with finitely many Conradian orderings. We show that, within this class of groups, having an isolated left-ordering is equivalent to having finitely many left-orderings.
In Chapter 1 we give the basic background and notations. We also give a new characterization of the Conrad property for orderings. In Chapter 2, we use the new characterization of the Conradian property to give a classification of groups admitting (only) finitely many Conradian orderings §2.1. Using this classification we deduce a structure theorem for the space of Conradian orderings §2.2. In addition, we are able to give a structure theorem for the space of left-orderings on a group by studying the possibility of approximating a given ordering by its conjugates §2.3. In Chapter 3 we show tha…
For any left orderable group G, we recall from work of McCleary that isolated points in the space of left orderings correspond to basic elements in the free lattice ordered group over G. We then establish a new connection between the kernels of certain maps in the free lattice ordered group over G, and the topology on the space of left orderings of G. This connection yields a simple proof that no left orderable group has countably infinitely many left orderings. When we take G to be the free group of rank n, this connection sheds new light on the space of left orderings of the free group: by a…
We show that no left-ordering on a free product of (left-orderable) groups is isolated. In particular, we show that the space of left-orderings of free product of finitely generated groups is homeomorphic to the Cantor set. With the same techniques, we also give a new and constructive proof of the fact that the natural conjugation action of the free group (on two or more generators) on its space of left-orderings has a dense orbit.
Let G be a countable group. We show there is a topological relationship between the space CO(G) of circular orders on G and the moduli space of actions of G on the circle; as well as an analogous relationship for spaces of left orders and actions on the line. In particular, we give a complete characterization of isolated left and circular orders in terms of strong rigidity of their induced actions of G on $S^1$ and R. As an application of our techniques, we give an explicit construction of infinitely many nonconjugate isolated points in the spaces CO(F_{2n}) of circular orders on free groups d…
We describe all possible ways of bi-ordering Thompson group F: its space of bi-orderings is made up of eight isolated points and four canonical copies of the Cantor set.
We describe a simple scheme for constructing finitely generated monoids in which left-divisibility is a linear ordering and for practically investigating these monoids. The approach is based on subword reversing, a general method of combinatorial group theory, and connected with Garside theory, here in a non-Noetherian context. As an application we describe several families of ordered groups whose space of left-invariant orderings has an isolated point, including torus knot groups and some of their amalgamated products.
We classify $C$-orderable groups admitting only finitely many $C$-orderings. We show that if a $C$-orderable group has infinitely many $C$-orderings, then it actually has uncountably many $C$-orderings, and none of these is isolated in the space of $C$-orderings. As a relevant example, we carefully study the case of Baumslag-Solitar's group B(1,2). We show that B(1,2) has four $C$-orderings, each of which is bi-invariant, but its space of left-orderings is homeomorphic to the Cantor set.
We study left orderable groups by using dynamical methods. We apply these techniques to study the space of orderings of these groups. We show for instance that for the case of (non-Abelian) free groups, this space is homeomorphic to the Cantor set. We also study the case of braid groups (for which the space of orderings has isolated points but contains homeomorphic copies of the Cantor set). To do this we introduce the notion of the Conradian soul of an order as the maximal subgroup which is convex and restricted to which the original ordering satisfies the so called conradian property, and we…
