What does it matter if a group has a non-elementary hyperbolic quotient?

My adviser recently shared a problem with me that seeks to establish non-elementary* hyperbolic quotients for mapping class groups. They told me that this could be useful for establishing results on separability or omnipotence, and that these could be relevant for examining profinite rigidity of hyperbolic 3-manifolds. Unfortunately, I'm not fully read up on these topics.

In this recent paper, Behrstock, Hagen, Martin and Sisto also seek to make headway on the question of hyperbolic quotients for mapping class groups. They have a discussion in their introduction of the relevance of this question, mentioning again separability and omnipotence, profinite rigidity, and placing things in the context of the virtual Haken conjecture. Again, I'm a little ignorant of these topics and their history of this point. So my question:

Q: Can anyone explain with some detail (or point me to some nice references as to) why it's relevant that mapping class groups have hyperbolic quotients? Or why it's helpful that any group has such a quotient?

*A Gromov-hyperbolic group is non-elementary if it is not virtually cyclic, i.e. is infinite and not virtually $$\mathbb{Z}$$.

• @dodd: on the contrary, as I explain in my answer, there is some precedent for using non-elementary hyperbolic quotients to study finite quotients. It's certainly difficult, but these are difficult problems,
– HJRW
Jan 21, 2021 at 22:11
• @dodd: Where...?
– HJRW
Jan 21, 2021 at 22:16
• @dodd: Since you don't point out where I assume hyperbolic groups are residually finite, I assume you concede that your previous comment was incorrect. On the last question we interacted on, you also made an incorrect comment and never conceded that you were wrong. This is not a good pattern of behaviour. There's no shame in being wrong.
– HJRW
Jan 22, 2021 at 7:46
• ... Regarding the "thinness" of the connection or otherwise, you are, of course, entitled to your opinion (although it doesn't carry much weight from an anonymous user), but it's irrelelant to the question, which asks for motivation to study hyperbolic quotients of mapping class groups. My answer lays out the motivation as I see it. I'd be very interested to hear anyone else's ideas.
– HJRW
Jan 22, 2021 at 8:31
• An infinite hyperbolic quotient is an obstruction to some strong versions of Property T, including V. Lafforgue's strong property T. I don't known if these properties are known to fail for MCGs.
– YCor
Jan 22, 2021 at 13:09

1 Answer

You don't get anything just from knowing that a non-elementary hyperbolic quotient exists. However, the problem of constructing such quotients of mapping class groups appears very difficult, and can be viewed as a step on the way to constructing other interesting classes of quotients that you might hope to study, like finite quotients or virtually abelian quotients.

For instance, here are two important open problems about the finite-index subgroups of mapping class groups (let's say for closed surfaces $$S$$ of genus $$\geq 3$$, to be safe). The first is easy to understand.

Virtual first betti numbers: Does $$\mathrm{Mod}(S)$$ have a finite index subgroup that surjects $$\mathbb{Z}$$?

The second is morally asking: "Does every finite-index subgroup of $$\mathrm{Mod}(S)$$ come from a finite-sheeted cover of $$S$$?"

Congruence subgroup property: Does every finite-index subgroup of $$\mathrm{Mod}(S)$$ contain a principal congruence subgroup, i.e. the kernel of a natural map $$\mathrm{Mod}(S)\to\mathrm{Out}(Q)$$, where $$Q$$ is a finite characteristic quotient of $$\pi_1(S)$$?

I'll take for granted that these are interesting questions, but could explain more if necessary. The second one is certainly related to the profinite rigidity questions mentioned in the question. (Added:) In any case, even if you don't like either of these questions specifically, the point is that we are interested in, and don't know much about, the finite and virtually abelian quotients of mapping class groups.

Note that either of these questions can be answered by finding suitable quotients of $$\mathrm{Mod}(S)$$: the first (positively) by finding a virtually abelian quotient, the second (negatively) by finding a finite quotient that doesn't factor through some $$\mathrm{Out}(Q)$$.

Neither can be answered directly by a non-elementary hyperbolic quotient. However, we know a lot about hyperbolic groups, and so if we can construct hyperbolic quotients, we might hope to go further and construct hyperbolic quotients with a rich supply of finite quotients (for instance if they're residually finite), and thereby answer these questions.

There's some precedent for this hope. The Virtual Haken conjecture and its cousins for hyperbolic 3-manifolds asked for rich supplies of finite quotients of hyperbolic 3-manifold groups. One of the crucial tools in Agol's proof turned out to be Wise's Malnormal special quotient theorem, which guarantees that hyperbolic 3-manifold groups have non-elementary hyperbolic quotients which are themselves residually finite. (This is a slightly backwards telling of the Virtual Haken story, but the point is that these are the tools you need.)

Indeed, in some ways the starting point for the proof of the Virtual Haken conjecture was a paper of Agol, Groves and Manning, which proved it (modulo the work of Kahn--Markovic) under the hypothesis that all hyperbolic groups are residually finite. I view the work of Behrstock, Hagen, Martin and Sisto as the analogue of the Agol--Groves--Manning paper in this story.

• The mapping class groups most probably have property (T) if the genus is large enough. This would kill question 1. Jan 21, 2021 at 22:12
• @dodd: As you say, that would answer question 1. But the jury is out as to whether it is "most probably" true. Indeed, some (admittedly, not many) people think the reverse has been proved. arxiv.org/abs/0706.2184
– HJRW
Jan 21, 2021 at 22:17
• That paper appeared 13 years ago, has never been published as far as I know and has been discussed here mathoverflow.net/questions/87310/…. On the other hand the paper arxiv.org/abs/1812.03456 will be soon (hopefully) published in the Annals of Mathematics. Jan 21, 2021 at 22:26
• @dodd: indeed. But it’s interesting that their computer programme hasn’t yet told us that any mapping class groups have property T. Regardless, the point is that the finite quotients of mapping class groups are interesting, and hyperbolic quotients suggest a means of studying them.
– HJRW
Jan 21, 2021 at 22:30
• @AGenevois: if you look at the comments on v2, you’ll see it has been retracted.
– HJRW
Jan 22, 2021 at 7:07