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Ordered Groups and Topology

Adam Clay and Dale Rolfsen
American Mathematical Society
Publication Date: 
Number of Pages: 
Graduate Studies in Mathematics 176
[Reviewed by
Michael Berg
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By definition (on p. 2 of the book under review) a group \(G\) is ordered iff \(G\) is a group and is totally ordered in both the left and the right invariant senses: if \(g<h\) in G, and \(f\) is any element of G, then \(fg<fh\) as well as \(gf<hf\). It is in fact the case (as the authors establish) that left invariance and right invariance are equivalent in the sense that a group can be given a left ordering iff it can be given a right ordering, so life is good. We’re dealing with one of those nice notions of, I guess, the intersection between group theory and universal algebra, and one reaps a lot of benefits accordingly. In any event, as Clay and Rolfsen indicate, there are lots of sources on ordered groups, but they “will focus mostly on groups of special topological interest and results relevant to topological applications.”

This is not to say that they sell ordered groups short. Quite the contrary: the book’s first three chapters develop a substantial amount of material on these objects and their behavior. Only thereafter do the topological themes take center stage. And these are very sexy themes indeed: knots, 3-manifolds, foliations, braid groups, and other such topics are covered with a good deal of zeal.

Here, to give a glimpse of what we’re dealing with are three representative results discussed in the book. First, regarding ordered groups as such we have Hölder’s Theorem (yes, it’s the same Hölder: the result dates to 1901): any group equipped with an ordering that is Archimedean is isomorphic to a subgroup of \(\mathbb{R}\). Here “Archimedean” means just what you’d think it means: given “positive” \(g,h\in G\) (yes, to the right of the group’s identity) there’s some natural number \(n\) such that \(h<g^n\). So intuition is given its due. Next, a result about knots: “Every knot group is locally indicable [see the book for the definition], and hence left-orderable.” It follows in particular that knot groups are torsion free. And finally, at the close of the chapter on foliations we find the following theorem: “Let M be a closed, irreducible, atoroidal rational homology 3-sphere that admits a co-orientable, taut foliation. Then the commutator subgroup [of M’s fundamental group] is left-orderable.” Hopefully these samples give at least some of the flavor of this book, and of its thrust given that it’s a huge understatement to say that knots are known by the groups they carry (and of course polynomials like Alexander’s or the HOMFLY).

Ordered Groups and Topology is offered as no. 176 in the AMS series, “Graduate Studies in Mathematics.” Given the huge popularity enjoyed by low dimensional topology these days, and all for good reason, it should make a very positive impact. The book is easy to read and deals with very pretty mathematics. I guess this is no surprise, given that Rolfsen (whose PhD student Clay is) is indeed the same Dale Rolfsen of Knots and Links, now published by AMS Chelsea, dating all the way back to 1976 and republished in 1990; it is interesting to read Lee Neuwirth’s enthusiastic Bulletin on the AMS review of that encyclopedic labor of love.

Three of my department colleagues are knot theorists (well, one of the trio is a recent convert from category theory: we’re pretty ecumenical): I might dangle this book in front of them just to tantalize them. Then again, maybe I’d better not…


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Michael Berg is Professor of Mathematics at Loyola Marymount University in Los Angeles, CA.

See the table of contents in the publisher's webpage.