Comments on: Surface subgroups of Sapir’s group

By: Danny Calegari

Danny Calegari — Sat, 22 Dec 2012 14:33:20 +0000

I get it – we have a surface fiber S, and cut it along a loop g, then pull one end of g once around the circle direction. Then one just needs to find an immersed subsurface of S bounding g-phi(g) where phi is the monodromy. Providing g-phi(g) is homologically trivial (eg if g is homologically trivial) such an immersed surface always exists, and can be chosen to represent all but finitely many classes in relative H_2. The surface obtained by gluing up is essential; actually, it’s even Thurston-norm minimizing, and in the boundary of the fibered face, since it’s transverse to the suspension flow (by max flow – min cut).

This is very interesting, since Alden and I actually do look at such surfaces in our paper – we call them “flat surfaces”, and they are pretty common, but not completely omnipresent, because the surface fiber S will have boundary that can’t be ignored so easily. We conjecture they can always be found virtually. But if one just lets them have boundary – so they are free groups rather than closed surface subgroups – they are indeed easy to find. Maybe there are enough to cubulate?

By: Henry Wilton

Henry Wilton — Thu, 08 Nov 2012 10:23:34 +0000

The idea is, in a fibred hyperbolic 3-manifold, to ‘cut and cross-join’ the fibre surface to produce immersed geometrically finite surfaces. The original construction was in C–L–R’s paper ‘Bundles and finite foliations’, I think. Masters showed that there are many such surfaces, in some sense, and Dufour showed that there really are enough to cubulate.

By: Danny Calegari

Danny Calegari — Wed, 07 Nov 2012 13:29:08 +0000

What is the C-L-R-M-D construction? What would be the analog in this context? What is codimension one about it?

By: Henry Wilton

Henry Wilton — Wed, 07 Nov 2012 12:51:19 +0000

The naive analogue of the Cooper–Long–Reid–Masters–Dufour construction (if it works!) would produce *free* subgroups of codimension one.

By: Danny Calegari

Danny Calegari — Wed, 07 Nov 2012 12:34:06 +0000

I don’t see a reason right now why these surface subgroups should be codimension one. But of course this is worth thinking about.

By: Henry Wilton

Henry Wilton — Wed, 07 Nov 2012 10:26:48 +0000

This is true (it would be a great theorem!), but there’s no particular reason to think that the codimension-one subgroups should be surface groups, is there?

By: Ian Agol

Ian Agol — Wed, 07 Nov 2012 04:13:32 +0000

It would be interesting if you could construct codimension-one quasiconvex subgroups of (hyperbolic) ascending HNN extensions of free groups. One might then be able to cubulate them this way. Guillaume Dufour did this in his thesis for mapping tori of surfaces (using a method of Cooper-Long-Reid and Masters to get quasiconvex surface subgroups).

By: Danny Calegari

Danny Calegari — Mon, 05 Nov 2012 23:57:03 +0000

Hi Ian – yes, I believe these surface subgroups should be quasiconvex (although we have not actually written down a proof). In any case, their injectivity is “certified locally”, and there should be some general principle that subgroups of hyperbolic groups whose injectivity is certified locally should be quasiconvex (although it is probably not easy to make that intuition precise).

If your toes send Henry an email, I’m sure he’ll be happy to fill them in.

By: Ian Agol

Ian Agol — Mon, 05 Nov 2012 22:31:16 +0000

Are these surface subgroups quasiconvex? And my toes are getting tired.

By: Henry Wilton

Henry Wilton — Mon, 05 Nov 2012 13:35:32 +0000

Thanks, Danny. I believe the proof of the new result is now correct and complete, but as it’s in somewhat unfamiliar territory for me I’d like to double-check it. So I would rather leave the multitudes waiting on tiptoes for the moment…

By: Danny Calegari

Danny Calegari — Mon, 05 Nov 2012 13:00:09 +0000

Hi Henry – thanks for the compliment, which I will happily accept on behalf of Alden and myself!

scl does make an appearance elsewhere in the paper, but proving that has a surface subgroup for random does not directly use scl (but you’re right that there is some overlap with Alden’s and my “Random rigidity” paper). However, a key step of the proof is computer assisted, and it turns out that scallop can be very easily adapted to look for f-folded surfaces just as effectively as for extremal surfaces.

By the way, I did think about mentioning your new result (which is obviously germane to the theme of this post), but I figured you might want to announce it yourself when you are ready. :)

By: Henry Wilton

Henry Wilton — Mon, 05 Nov 2012 10:18:13 +0000

Well done! This is a really great result.

I notice that you don’t mention scl in this post. Does it appear explicitly in the details of the proof, or is its influence here just implicit? (For instance, the ideas here seem related to your joint proof that a random endomorphism is an scl-isometry.)