Dani Wise gave a series of lectures at the Wasatch Topology Conference back in December about his very exciting work on quasi-convex hierarchies and residual properties of groups (like subgroup separability). Thomas Koberda took detailed notes which he polished into a useful 9 page summary of Dani’s work. (I’d been thinking I should blog about Dani’s talks, which I also attended, and I’m very glad that Thomas saved me the trouble…)
January 13, 2010
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Low Dimensional Topology 
Dani Wise has a research announcement:
http://www.aimsciences.org/journals/displayArticlesERA.jsp?paperID=4703
It has only a bit more information than Thomas Koberda’s notes.
Steve Boyer and I are organizing a workshop in April on the virtual Haken
conjecture and related questions. Before the workshop, Dani Wise is going to
give a series of talks on his work.
http://www.cirget.uqam.ca/3manifolds/index_e.shtml
Comment by Ian Agol — January 19, 2010 @ 6:06 pm |
In the August 31, 2009 post on the Virtual Haken Conjecture you stated that “Wise claims a proof of Conjecture 7 in the special case that M is a Haken manifold containing an embedded quasi-fuchsian surface”. From my understanding, the result presented at the Wasatch Topology Conference back in December is that Conjecture 6 is true for M a Haken manifold and he does not need the hypothesis of an embedded quasi-fuchsian surface. Is that correct? How does his current result relate to Conjecture 7? Does he still need the assumption of an embedded quasi-fuchsian surface for Conjecture 7?
Comment by Mayer A. Landau — January 21, 2010 @ 8:41 am |
You’re correct that Wise does not need the assumption of a quasi-fuchsian surface in his announced proof of Conjecture 6. However, he does definitely need it for Conjecture 7 . The reason for this is if M is a Haken 3-manifold, then by Thurston’s dichotomy the incompressible surface it contains is either a fiber in a fibration over the circle (*) (and so there’s nothing to show) or is quasi-fuchsian (in which case apply Wise’s proof of Conjecture 7 together with the work of Ian Agol).
(*) To be precise, I should say that becomes the fiber in such a fibration in a two-fold cover of M.
Comment by Nathan Dunfield — January 21, 2010 @ 10:10 am |
Well… if the fibration case is LERF, and the quasi-fuchsian case is LERF, and by Thurston’s dichotomy there are only these two possibilities, then doesn’t that imply that all hyperbolic Haken manifolds are LERF?
Comment by Mayer A. Landau — January 21, 2010 @ 8:31 pm |
The issue is that the fibration case is currently not known to be LERF, except in certain cases (e.g. the first betti number is greater than 1, or if the fiber has genus 2). The conclusion to Conjecture 6 is that the manifold virtually fibers, not that it is LERF.
Comment by Nathan Dunfield — January 21, 2010 @ 9:43 pm |
Nathan, when you say that LERF is known when the Betti number is greater than 1, do you mean that that case follows from Wise’s work? Or is it known for some other reason?
Comment by Henry Wilton — January 26, 2010 @ 8:58 pm
Yes, I mean it follows from Wise’s work. Thurston showed, via his eponymous norm, that if
then there is always a surface which is not a fiber, hence quasifuchsian, hence Wise applies. Specifically, anything not in the interior of a top-dimensional face of the Thurston norm ball is a non-fiber.
Comment by Nathan Dunfield — January 27, 2010 @ 8:24 am
I see, when you said there was nothing to show, you were referring to Conjecture 6.
Comment by Mayer A. Landau — January 21, 2010 @ 9:45 pm |
Exactly.
Comment by Nathan Dunfield — January 22, 2010 @ 8:36 am |