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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.

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    $\begingroup$ 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. $\endgroup$
    – Joel David Hamkins
    Commented Apr 18, 2016 at 16:07
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    $\begingroup$ 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. $\endgroup$
    – Douglas Zare
    Commented Apr 19, 2016 at 8:26
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    $\begingroup$ @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... $\endgroup$
    – Mikhail Katz
    Commented Apr 19, 2016 at 8:33
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    $\begingroup$ @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? $\endgroup$
    – Douglas Zare
    Commented Apr 22, 2016 at 5:32
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    $\begingroup$ @MikhailKatz: Have you read the abstract? $\endgroup$
    – Douglas Zare
    Commented May 1, 2016 at 19:52

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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.

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  • $\begingroup$ I found Mj only as a moderator, not a speaker, at the 2022 ICM. Do you have a link? $\endgroup$
    – LSpice
    Commented Aug 8, 2022 at 21:52
  • $\begingroup$ Perhaps this is the intended talk (at ICM 2018)? youtube.com/watch?v=BGk2FYiHEhk $\endgroup$
    – aorq
    Commented Aug 8, 2022 at 23:49
  • $\begingroup$ My mistake - I will fix the reference to Mahan Mj and will add the link. $\endgroup$
    – Sam Nead
    Commented Aug 9, 2022 at 7:07
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    $\begingroup$ 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. $\endgroup$
    – Sam Nead
    Commented Aug 9, 2022 at 9:26
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    $\begingroup$ 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. $\endgroup$
    – Emil Jeřábek
    Commented Aug 9, 2022 at 9:52

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