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?

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

]]>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. :)

]]>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.)

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