Patrick Dehornoy, Ivan Dynnikov, Dale Rolfsen, and Bert Wiest

Abstract: In the decade since the discovery that Artin's braid groups enjoy a left-invariant linear ordering, several quite different methods have been applied to understand this phenomenon. This book is an account of those techniques, including self-distributive algebra, finite trees, combinatorial group theory, mapping class groups, laminations and hyperbolic geometry.

Soc. Math. France series Panoramas et Syntheses 14(2002). For further details click HERE

Although you are strongly encouraged to buy the book (it is moderately priced), a PDF version is available HERE Warning: it is a large file, approximately 200 pages!

Dale Rolfsen

A new, revised edition is published by AMS Chelsea Press, 2003. CLICK HERE FOR MORE INFORMATION

Dale Rolfsen

These are notes from a one-hour lecture.

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Dale Rolfsen

These are notes for a minicourse on the braid groups at the Institute for the Mathematical Sciences, National University of Singapore

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Dale Rolfsen

These are notes adapted from slides used in a minicourse on Ordered Groups and Topology, consisting of three 75-minute lectures at CIRM, Luminy, in June 2001. They are quite sketchy, with few proofs, as they were just slides. Eventually I will expand these notes into a monograph.

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Dale Rolfsen

These are slides from a lecture on various conjectures related to the Poincare conjecture.

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Adam Clay and Dale Rolfsen

We establish a necessary condition that an automorphism of a nontrivial Þnitely gener- ated bi-orderable group can preserve a bi-ordering: at least one of its eigenvalues, suitably deÞned, must be real and positive. Applications are given to knot theory, spaces which Þbre over the circle and to the HeegaardÐFloer homology of surgery manifolds. In particular, we show that if a nontrivial Þbred knot has bi-orderable knot group, then its Alexander polyno- mial has a positive real root. This implies that many speciÞc knot groups are not bi-orderable. We also show that if the group of a nontrivial knot is bi-orderable, surgery on the knot cannot produce an L -space, as deÞned by Ozsvath and Szabo.

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Peter A. Linnell, Akbar H. Rhemtulla and Dale Rolfsen

This paper investigates conditions under which a given automorphism of a residually torsion-free nilpotent group respects some ordering of the group. For free groups and surface groups, this has relevance to ordering the fundamental groups of three-dimensional manifolds which fibre over the circle.

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Bernard Perron and Dale Rolfsen

This extends our results of a previous paper to 3-manifolds which fibre over the circle, and have closed fibres. If a 3-manifold M fibres over the circle, with oriented fibre F, then the fundamental group of M is bi-orderable if the homology monodromy has all eigenvalues real and positive.

Math. Proc. Camb. Phil. Soc. (to appear 2006).

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Dale Rolfsen

This is based on a lecture given at a NATO-PIMS Conference on Quantum Topology, in Alberta, 2001. It describes an obstruction to finding a map of nonzero degree between two given 3-manifolds, based on orderability of their groups. The theoretical aspects of this paper are from the paper by Boyer, myself and Wiest, listed below, but this paper expands the result with many examples.

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Steve Boyer, Dale Rolfsen and Bert Wiest

Surprisingly many fundamental groups of 3-manifolds can be ordered in such a way that the ordering is invariant under multiplication on one, or even both, sides. We characterize exactly which Seifert fibred spaces have such groups. The relationship between such orderings and fibrations, foliations and other geometric structures is explored. A sample result is that there exist manifolds in each of the eight Thurston geometries which have orderable groups, and there also exist examples in each which do not. Orderability is also closely related to the existence of taut foliations and laminations, the virtual Haken conjecture and other important questions regarding 3-manifolds.

Ann. Inst. Fourier 55(2005), 243-288.

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Bernard Perron and Dale Rolfsen

All classical knot groups can be ordered in a left-invariant manner and many of them, such as torus knots, cannot be bi-invariantly ordered. We show that the group of a fibred knot, whose Alexander polynomial has all roots real and positive, can be bi-ordered. The simplest example is the figure-of-eight knot.

Proc. Cambridge Phil. Soc. 135(2003), 147-153.

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Djun Kim and Dale Rolfsen

Canadian J. Math. 55(2002), 822-838.

An explicit ordering of the pure braid groups is constructed, which is invariant under multiplication on both sides. This has the special property that, restricted to pure braids which are "positive" in the sense of Garside, it is a well-ordering.

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Akbar Rhemtulla and Dale Rolfsen

Proc. Amer. Math. Soc. 130(2002), 2569-2577 (electronic).

Locally indicable groups can be given an ordering which is invariant under right multiplication, but not conversely. We characterize those right-orderable which are locally indicable. This was motivated by a problem in ordering braid groups. The braid groups are known to be right-orderable, whereas the pure braid groups have a 2-sided ordering. We show that such orderings are necessarily incompatible. We also give a new proof of the result of P. Linnell, that for elementary amenable groups, local indicability and right-orderablility coincide.

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Dale Rolfsen and Bert Wiest

Algebraic Geom. Topology 1(2001), 311-320 (electronic)

Free groups have orderings which are invariant under multiplication on both sides. We investigate under which conditions such an ordering can be found which is, in addition, invariant under certain families of automorphisms. As an application, we show that all surface groups have two-sided invariant orderings, with the sole exceptions of the projective plane and Klein bottle. This corrects an error in the literature, since 1942, which asserted that fundamental groups of nonorientable surfaces cannot have bi-invariant orderings.

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Luis Paris and Dale Rolfsen

Jour. Reine Angew. Math. 521(2000), 47-83

Inclusions of Riemann surfaces (boundaries and marked points allowed) induce homomorphisms of their respective mapping class groups, which are typically (but not always) injective. The subgroup corresponding to a subsurface in this way is called a geometric subgroup. We study these geometric subgroups and calculate their normalizers, commensurators, etc. The techniques are elementary and the paper is essentially self-contained.

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Dale Rolfsen

A survey paper presented at the Conference on Quantum Invariants of 3-manifolds in Alberta, 2000. It discusses some older results, not so well-known, as well as new results such as left-orderability and linearity of the braid groups.

Topology and Applications 127(2003), 77-90.

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Luis Paris and Dale Rolfsen

Annals Inst. Fourier 49(1999), 417-472

To any Riemann surface and positive integer n, there is an associated surface braid group, called the n-strand braid group of the surface. Inclusion of a surface into a larger surface induces homomorphisms of their various braid groups. We study when these are injective or not, and calculate the centralizers, normalizers, etc. of these geometrically defined subgroups of the surface braid groups.

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David Austin and Dale Rolfsen

Canadian Math. Bull. 42(1999), 257-262

It is shown that any knot in a homology 3-sphere is homotopic to one with trivial Alexander polynomial. This has applications to an inductive skein-theoretic definition of SU(2)-signatures of knots in homology spheres.

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Dale Rolfsen and Jun Zhu

J. Knot Theory and Ramifications 7(1998), 837-841

We point out that Dehornoy's recent discovery that the braid groups are left-orderable implies that the group algebras of these groups have no zero divisors and (hence) no idempotents, a previously unkown result. We also argue that the pure braid groups have an ordering invariant under multiplication on both sides, using results of Falk and Randall.

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Dale Rolfsen

Invent. Math. 130(1997), 575-587

The normalisers and commensurators of certain subgroups of the braid groups,
B(n), are calculated. In particular this is done for the naturally included
subgroups B(k), with k