Low Dimensional Topology

June 21, 2011

Wise’s work on groups with quasiconvex hierarchies (again)

Filed under: Geometric Group Theory,Hyperbolic geometry,Virtual Haken Conjecture — Nathan Dunfield @ 5:10 pm

This blog has mentioned several times Dani Wise’s work on subgroup separability properties for certain word-hyperbolic groups [1, 2, 3]. In August, there will be a CBMS-NSF conference at CUNY focusing on this work, and the reason for this post is that one major part of Dani’s work is now available on the conference website.

(Thanks to Jason Behrstock and Ian Agol for telling me that the preprint had been posted there.)



  1. Why do you say “one major part”. Is there more? Corollary 14.3 has the virtually fibered result, and what appears to be a the LERF result spoken of in previous posts.

    Comment by Mayer A. Landau — June 23, 2011 @ 1:17 am | Reply

    • There are two other main new pieces that are used in the proof of Corollary 14.3:

      [HWa] Frederic Haglund and Daniel T. Wise. A combination theorem for special cube complexes. Submitted.
      [HWc] Tim Hsu and Daniel T. Wise. Cubulating malnormal amalgams. Preprint.

      The first is available online, but I don’t think the second paper is.

      Roughly, the the above two papers deal with the case when one is gluing along malnormal subgroups, and the paper the post is about is about ways to reduce the general case to the malnormal one.

      The paper [HWa] is a generalization of Wise’s Inventiones paper about amalgams of free groups.

      Comment by Nathan Dunfield — June 23, 2011 @ 9:50 am | Reply

  2. One more question. In the January blog post you stated
    “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 .”
    But in this preprint, in Corollary 14.3 Wise states \pi_{1}M is subgroup separable, i.e. LERF.
    There seems to be no quasi-fuchsian hypothesis in this corollary. Does geometrically finite in this context imply quasi-fuchsian?

    Comment by Mayer A. Landau — June 24, 2011 @ 8:56 am | Reply

    • Yes, in this context geometrically finite and quasi-fuchsian are equivalent (this was shown by Bers in the 1970s, I think).

      Comment by Nathan Dunfield — June 24, 2011 @ 9:01 am | Reply

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