Adam Clay

Graph manifolds, left-orderability and amalgamation

We show that every irreducible toroidal integer homology sphere graph manifold has a left-orderable fundamental group. This is established by way of a specialization of a result due to Bludov and Glass [Proc. Lond. Math. Soc. 99 (2009) 585–608] for the amalgamated products that arise, and in this setting work of Boyer, Rolfsen and Wiest [Ann. Inst. Fourier (Grenoble) 55 (2005) 243–288] may be applied. Our result then depends on known relations between the topology of Seifert fibred spaces and the orderability of their fundamental groups.

A new characterization of Conrad’s property for group orderings, with applications

We provide a pure algebraic version of the dynamical characterization of Conrads property. This approach allows dealing with general group actions on totally ordered spaces. As an application, we give a new and somehow constructive proof of a theorem first established by Linnell: an orderable group having infinitely many orderings has uncountably many. This proof is achieved by extending to uncountable orderable groups a result about orderings which may be approximated by their conjugates. This last result is illustrated by an example of an exotic ordering on the free group given in the Appendix.

Testing bi-orderability of knot groups

We investigate the bi-orderability of two-bridge knot groups and the groups of knots with 12 or fewer crossings by applying recent theorems of Chiswell, Glass and Wilson. Amongst all knots with 12 or fewer crossings (of which there are 2977), previous theorems were only able to determine bi-orderability of 599 of the corresponding knot groups. With our methods we are able to deal with 191 more.

Isolated points in the space of left orderings of a group

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 guarantee that it is not isolated in LO(G), and we illustrate these ideas using the Dehornoy ordering of the braid groups.

DENSELY ORDERED BRAID SUBGROUPS

Dehornoy showed that the Artin braid groups Bn 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 Bn which is discretely ordered by the Dehornoy ordering.

EXOTIC LEFT-ORDERINGS OF THE FREE GROUPS FROM THE DEHORNOY ORDERING

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.

Corrigendum to: "On ordering free groups" [J. Symbolic Comput. 40 (2005) 1285-1290]

The maritime pump of present invention has a plurality of crutches and a floating block which keeps the maritime pump at the proper position in seawater. The crutches are set at the lateral sides of the pump body, which stabilize the pump body and make the pump body slide along with it. The density of the aforementioned floating block is lower than seawater and may be put on the top or the bottom of the pump body so that it keeps the pump body at the proper position of seawater. The pump body is a tube-shaped object and is attached with a hopper, to lead seawater into the pump body. The shape and size of the piston are the same as the area of the space of the tube-like pump body. The piston has guiding holes in the relative positions of the guiding sticks, so that the guiding sticks can penetrate the guiding holes and make the piston slide along with the guiding sticks to push seawater.

Left orderings and quotients of the braid groups

In this note, we show that a homomorphism from the braid group Bn into an arbitrary group is injective if and only if its image can be left ordered in a particular way. It follows from this criterion that the image of Bn (n ≥ 2) under any homomorphism is bi-orderable if and only if it is infinite cyclic. As a second application, this criterion also provides a new proof that the Artin representation of Bn into Aut(Fn) is injective.