Thanks for pointing this out, Ian. Incidentally, as far as I can tell, if Gromov conjectured anything, it was that there are one-ended word-hyperbolic groups *without* surface subgroups (see p. 144 of ‘Hyperbolic Groups’, *Essays in group theory*…). Of course, it would be interesting to know if he had conjectured something else somewhere else.

Vlad has written a nice paper explaining how if one could construct “enough” quasi-convex surface subgroups of a hyperbolic group with 2-sphere ideal boundary, then one could prove it is virtually a Kleinian group. Enough means for any pair of points on the sphere, there is a circle limit set of a quasi-convex surface subgroup which separates the two points. http://front.math.ucdavis.edu/1205.5747

Of course, proving the existence of quasi-convex surface groups still is quite non-trivial. In Kahn-Markovic’s construction, they make use of some delicate geometric estimates to get the surfaces to be nearly geodesic, and they make use of the strong mixing of the frame flow in a hyperbolic manifold, which is a result due to Moore and makes use of representation theory of SL(2,C). So it’s not yet clear whether this will be a viable approach to the Cannon conjecture. But hopefully it will encourage people to think more about Gromov’s conjecture about surface subgroups of hyperbolic groups. Note that a result of Kapovich-Kleiner would imply that hyperbolic groups with Sierpinski carpet ideal boundary would be Kleinian groups if Cannon’s conjecture is true. http://www.sciencedirect.com/science/article/pii/S0012959300010491

]]>@ Henry (I’m not sure why I can’t reply to your comment, so I’ll reply to mine!): The idea would be to pass to a cover in which all of the surfaces embed and are acylindrical (i.e. pass to a subgroup in which all of the surface subgroups are malnormal). Then you create a hierarchy inductively out of the embedded surfaces, cutting along each one inductively. Two embedded surfaces intersect in a subsurface, so one uses pieces of the complements of these subsurfaces to create the hierarchy. So it doesn’t actually give a 3-dimensional cubulation, even of a finite-sheeted cover. In any case, I should attribute this approach to Markovich. When I had thought of this originally, I was contemplating using some sort of abstract hierarchy coming from a (high-dim) cubulation, and carrying out Thurston’s geometrization of Haken manifolds using the skinning map etc. But Vlad pointed out to me that one need only show that a finite-index subgroup extends as a group action over the ball.

]]>That’s very interesting. Is it possible to say a bit more about how the hierarchy helps one to extend the action over the ball? It seems germane to point out that Tao Li gave examples of 3-manifolds that don’t admit 3-dimensional cubulations (as I think I learned in a talk that you gave, Ian!).

]]>There’s some hope that one can perform the Kahn-Markovic construction for hyperbolic PD3 groups to show that they are cubulated. Then virtual specialness should imply that they are 3-manifold groups, by extending the action over the ball in a finite-sheeted cover using the hierarchy. Markovic pointed out to me that one also obtains another proof of Tukia’s conjecture (proved by Casson-Jungreis and Gabai) since you can cubulate a hyperbolic group with boundary a circle by cyclic subgroups.

]]>Peter, the Scott core theorem doesn’t play any role in this as far as I can tell. Instead, the proof crucially relies on the group being Gromov hyperbolic and having a ton of “codimension 1” subgroups (which in the case of hyperbolic 3-manifolds come from Kahn-Markovic). I think Cannon conjectured that a hyperbolic PD3-group is in fact the fundamental group of a hyperbolic 3-manifold, but I guess that’s a separate issue.

]]>If you think about the finite case, the existence of such groups is fairly obvious. For instance, the cyclic group of order two is a subgroup, and hence a subgroup of finite index, of .

]]>Yes, that is exactly why I was puzzled, since as you yourself pointed out, most hyperbolic 3-manifolds are homology spheres. I did not realize you could have a perfect group with a non-perfect finite index subgroup.

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