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This question arose in the context of an earlier question on Grothendieck's critique of the traditional foundations of topology. Can the paper Group Invariant Peano Curves by Cannon and Thurston be regarded as exhibiting the naturality of phenomena like space-filling curves previously regarded as odd?

To clarify, the notion of naturality is not mine but rather is implied by Grothendieck's critique of the foundations of topology; see the earlier question linked above (perhaps he even uses the term natural, I would have to double-check). According to that view, odd phenomena like space-fillling curves are an artifice of the foundations. The question is whether the Cannon-Thurston paper finds them to be actually useful for something, which may be an argument against Grotendieck's criticism.

Sam Nead
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Mikhail Katz
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    Did you have a mathematically precise concept of "natural" in mind? I think there is no universally agreed-upon concept of what counts as natural. – Joel David Hamkins Apr 18 '16 at 16:07
  • "...regarded as exhibiting the naturality of phenomena..." ? – Włodzimierz Holsztyński Apr 18 '16 at 17:57
  • See my clarification in the question. – Mikhail Katz Apr 19 '16 at 07:36
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    Yes, space-filling curves appear in the wild, even though they may seem to be contrived at first. There is a natural $S^1$ at infinity for a hyperbolic surface, and a natural $S^2$ at infinity for a hyperbolic $3$-manifold, and it might be that when you have a surface in a $3$-manifold that there is a well-defined map from the $S^1$ at infinity to the $S^2$ at infinity that is continuous and onto. The Hopf fibration is also natural even if it seems strange at first. – Douglas Zare Apr 19 '16 at 08:26
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    @Douglas, note that for Fuchsian groups (a.k.a. surfaces) the circle at infinity is pretty tame and it is only when you start looking at subgroups of Kleinian groups that you get wilder behavior for the limit set. Also notice that these are not space-filling curves since they are still homeomorphic to a circle. I personally have nothing against space-filling curves but I am just wondering if this is because of their intrinsic merit or because of what Grothendieck describes as... – Mikhail Katz Apr 19 '16 at 08:33
  • ..."the almost insurmountable inertia of the mind, burdened by a heavy weight of conditioning, which makes it difficult to take a real look at a foundational question, thus at the context in which we live, breathe, work – accepting it, rather, as immutable data." – Mikhail Katz Apr 19 '16 at 08:34
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    In my understanding, Grothendieck does not say that wild phenomena do not exist, but that an inadequate formalism diverts our attention from our initial problems to problems which are irrelevant to the given situation (he mentions Brouwer's invariance of the domain as an example). Also, Grothendieck insists that the category of good spaces should allow for reasonable constructions. He also mentions that these good spaces should have triangulation. (All of this is realized by o-minimal geometries.) ... – ACL Apr 19 '16 at 16:58
  • @ACL, I agree. My understanding is that he is saying that things like invariance of domain should be easy consequences once the "right" definitions are in place. – Mikhail Katz Apr 19 '16 at 17:04
  • ... so my interpretation of the Cannon/Thurston paper you mention is that such objects are not tame in Grothendieck's sense. However, I do not know whether the problem lies in the specific hyperbolic 3-manifold fibered over the circle or in its universal covering... – ACL Apr 19 '16 at 17:05
  • I can't really comment intelligently on Cannon-Thurston since I haven't read it and nobody bothered to summarize it yet. – Mikhail Katz Apr 19 '16 at 17:06
  • NB. O-minimal geometries have cellular decompositions and a good theory of dimension, so that invariance of domain is obvious (once the machinery is set up). A natural theory of dimension is of course prevented by the existence of space-filling curves. – ACL Apr 19 '16 at 17:06
  • @ACL, I think you have a fine answer to my question posted here. Could you contribute something there? – Mikhail Katz Apr 20 '16 at 07:23
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    @Mikhail Katz, I don't understand why you were so dismissive of my comment. I tried to explain how space filling curves arise naturally in the Cannon and Thurston paper for non-topologists, and you say "no one has bothered to summarize it..." If you want to throw out these space-filling curves that arise from important structures (surface bundles over circles) that were defined earlier, what part of mathematics do you need to destroy? Can we not look at hyperbolic structures any more, since they might produce space filling curves? – Douglas Zare Apr 22 '16 at 05:32
  • @DouglasZare, I appreciated your comment. I was merely pointing out that I was not familiar with the Cannon-Thurston paper. In fact I am still not familiar with it, so it is hard to comment meaningfully on it. – Mikhail Katz May 01 '16 at 15:13
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    @MikhailKatz: Have you read the abstract? – Douglas Zare May 01 '16 at 19:52

1 Answers1

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Here I am expanding Doug Zare's comment into an answer.

The original poster asks:

Can the paper Group Invariant Peano Curves by Cannon and Thurston be regarded as exhibiting the naturality of phenomena like space-filling curves previously regarded as odd?

and

The question is whether the Cannon-Thurston paper finds them to be actually useful for something, which may be an argument against Grothendieck's criticism.

I do not have to hand Grothendieck's definition of "natural" nor the poster's definition of "useful for something", so I will substitute my own.

I take "naturality" to mean "without making many choices" and thus "having unexpectedly good properties". For the various Peano curves, many choices are needed. There are more-or-less simple recursive constructions, but this only underlines the fact that you could bolt these together in various ways to get uncountably many examples.

On the other hand, to obtain a Cannon-Thurston map, only two choices are required: first a fibered hyperbolic three-manifold $M$ and second a fibering of $M$.

With the real choices out of the way, the construction of the CT map is as follows. We choose (but this does effect the eventual answer) an elevation of $F$ - that is, an embedding of the universal cover of $F$ into the universal cover of $M$. The former is a copy of $\mathbb{H}^2$ while the latter is a copy of $\mathbb{H}^3$. Any hyperbolic space $\mathbb{H}^n$ has a boundary at infinity - this is a copy of the sphere $S^{n-1}$. Thus we obtain a map $\phi \colon S^1 \to S^2$. That this map is continuous requires work. That it is sphere-filling (and finite-to-one, and has certain invariance properties) is easier.

Note that if $N \to M$ is a finite cover, then the fibering of $M$ induces one of $N$. Let $G$ be the fiber in $N$. Thus there is a covering map $G \to F$. The CT maps associated to $(M, F)$ and $(N, G)$ are then isomorphic, and there is a diagram expressing the commutivity.

This is also useful for various things. For example, Mahan Mj has used the existence of (more general) CT maps to prove that the limit set of a (finitely generated) Kleinian group is locally connected; see his talk at the 2018 ICM.

Sam Nead
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  • I found Mj only as a moderator, not a speaker, at the 2022 ICM. Do you have a link? – LSpice Aug 08 '22 at 21:52
  • Perhaps this is the intended talk (at ICM 2018)? https://www.youtube.com/watch?v=BGk2FYiHEhk – aorq Aug 08 '22 at 23:49
  • My mistake - I will fix the reference to Mahan Mj and will add the link. – Sam Nead Aug 09 '22 at 07:07
  • I’m curious. How does one pronounce Mj? Is it an abbreviation or an actual name? – Emil Jeřábek Aug 09 '22 at 08:35
  • It is an abbreviation - sort of like how "Dr" stands for "Doctor". His "actual" name is Mahan Mitra, but he no longer uses it, I think. Ah, he has a wikipedia page -- https://en.wikipedia.org/wiki/Mahan_Mj -- so you can find references there. (Oh, and I would just say "mij". Or just "Hey, Mahan, good to see you again!") – Sam Nead Aug 09 '22 at 08:46
  • The wikipedia page was the first place I looked at, and found no explanation there. So what does Mj stand for? – Emil Jeřábek Aug 09 '22 at 09:00
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    The second line of the wikipedia page says that his name is also "Mahan Maharaj". So perhaps Mj is a shortening of "Maharaj". However, I vaguely remember that him mentioning a much longer title? I am afraid that if you want a definitive answer you'll have to ask Mahan. – Sam Nead Aug 09 '22 at 09:26
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    All right, thank you. Yes, it did occur to me that it might stand for Maharaj, but then trying to confirm it with another search, I came out empty-handed. That's why I decided to ask here, thinking that someone may know it first-hand. – Emil Jeřábek Aug 09 '22 at 09:52