Found! (very probably…)

I think I’ve found the mysterious author of the notes on 3-manifolds: it is (or should be) G. P. Scott. The crucial clue is the fact that the notes claim that the author, and Shalen independently, proved that “3-manifold groups are coherent”, and then gives the proof. This would immediately clarify things, were it not for the fact that (1) Shalen never published his proof; (2) the terminology “coherent” doesn’t seem to be really well known for groups, really. What it is defined to mean is the following: a group G is coherent if and only if, all its finitely generated subgroups are finitely presented.

But, as it happens, even Scott’s paper proving this doesn’t seem to use the terminology! (In MathSciNet, there are ten papers by someone named Scott including “coherent” somewhere in the review — but again that one is not among them) Fortunately, Google did find some references for “Shalen coherent”, in particular a Bourbaki seminar by J. Stallings reporting on Scott’s result (which gives, in particular, simple examples of non-coherent groups).

[Note: On Scott’s page, I found what seems to be a quite nice survey of The geometries of 3-manifolds, from 1983.]

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Kowalski

I am a professor of mathematics at ETH Zürich since 2008.

5 thoughts on “Found! (very probably…)”

  1. I saw this post too late to be of any use in your detective work (which in any case was apparently successful). Scott has been writing a book on 3-manifolds for something like 20 years (maybe more) – I wonder if this is an early draft?

    By the way, the use of “coherent” (to mean that all finitely generated subgroups are finitely presented) is standard terminology amongst geometric group theorists (see eg http://www.jstor.org/pss/121081 ); nevertheless I tend to forget what it means and have to keep asking every time it comes up.

    Probably the reason that “Scott” and “coherent” don’t show up together via Google is that Scott’s theorem and proof are very *geometric*: he shows that every 3-manifold with finitely generated fundamental group is homotopic to a compact submanifold (a “core”). This has as a consequence the corollary that 3-manifold groups are coherent, but the result is more often quoted and used in the language of manifolds.

  2. In fact, a preprint version of Scott’s book from 2003 on page 3 has the following “In the case of dimension five or more, we know from the work of Kirby and Siebenmann [??] that the PL and TOP classifications of manifolds are different. In the case of dimension seven or more, we know that the PL and DIFF classifications are also different.” which exactly matches the first half of paragraph 2 of the typewritten notes (the 2003 draft diverges before and after, but has parts which match better or worse). Do you have any idea of a date for the typewritten notes? Do the references give a lower bound?

  3. The date must be around 1974; it cites as “to appear” the following paper:

    Swarup, G. Ananda On incompressible surfaces in the complements of knots. J. Indian Math. Soc. (N.S.) 37 (1973), 9–24 (1974).

    Amusingly, Scott’s paper (from 1973) is also cited, but not referred-to at the beginning of the section where he says “This result was proved by myself and by Shalen independently”…

    Once I get back to Switzerland, I may use a PDF-producing xerox machine to create a PDF version.

  4. I talked to Peter, and he recalls that these are for some lectures he gave at Maryland in December of 1973. No idea how they came to be where you found them.

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