Thurston-Cannon $S^2$-filling curves

Question

I have been looking into equivariant space-filling curves $S^1\to S^2$ as discussed by Cannon & Thurston in these two papers:

• Three-Dimensional Manifolds, Kleinian Groups and Hyperbolic Geometry
• Group-Invariant Peano Curves

The title of the second paper suggests a resemblance to the Peano Curve

The Cannon-Thurston construction seems a little more delicate. In the first paper, near Theorem 5.7 we get

• Let $M_\phi$ be the mapping torus of a (pseudo-Anosov map) $\phi: S \to S$. This is $S \times [0,1]$ with the ends identified by $\phi$: $(S,0)\sim_\phi (S,1)$.
• We can extend this map to the universal cover and get a map from the hyperbolic plane to hyperbolic space $\tilde{\phi}: \mathbb{H}^2 \to \mathbb{H}^3$.
• For reasons I do not fully undestand, they ask if this extends to a map from $S^2 \to S^3$ which involves checking of the map exists $S^1_\infty \to S^2_\infty$ "near infinity".

Thus we get a sphere-filling curve. The images supplied in Thurston's paper do not convince me they fill the sphere.

Nor does this picture look very sphere-filling.

In what sense does the figure about fill the sphere and how is this related to Peano space-filling curve of $[0,1]^2$ ?

Answer 1

If you'd like to see a more explicit picture of these Peano curves in the case of cusped manifolds, check out the paper of Alperin, Dicks, and Porti (see also subsequent papers of Cannon and Dicks). They give a subdivision tiling in figures 16 and 17 of the paper which shows a better analogy with the Peano curve.

The picture in Figure 10 is supposed to indicate the points that are identified in the map from the circle to $S^2$. Such a map cannot be one-to-one; however it is finite-to-one, and the points that are identified form the endpoints of the leaves of the stable and unstable laminations of the pseudo-Anosov map.

I'm not certain how Figure 12 was drawn, but I suspect it's a picture of a geometrically finite punctured torus group, which is "close" to being degenerate, but isn't actually, which might account for why it doesn't appear to be sphere filling.

Answer 2

There are multiple reasons that the Cannon-Thurston curves are related to the Peano curve. For instance, these are all continuous surjective paths onto 2-dimensional objects: for the original Peano curve the target is the square; for the Cannon-Thurston curves the target is the 2-sphere.

Also, the original Peano curve is constructed as a limit of a sequence of piecewise linear paths. If one thinks about the original Peano construction, one sees that the paths in the sequence can all be obtained from the final Peano curve by "connecting dots", that is to say by taking a finite sequence of points along the Peano curve and interpolating by straight line segments. The pictures of Cannon-Thurston curves you have exhibited were (I am pretty sure), also constructed by "connecting the dots", yielding some single term of an approximating sequence of paths.

As with any sequence, even say a sequence of rational numbers converging to a real number, there are important issues of convergence rate. "Connect-the-dots" approximations of Cannon-Thurston Peano curves converge rather rapidly for noncusped fibered 3-manifolds. But for cusped fibered manifolds they converge quite slowly. In Figure 12, right in the center is a cusp point, and the curve is actually differentiable exactly at the cusp point. The algorithm for drawing the connect-the-dots approximation is wasting lots of time near that cusp point (and others). Hence the lack of time to spend drawing any part of the curve that fills in the white spaces.