Saul Schleimer, Henry Segerman, Will Segerman

The white and blue regions of this carving have a common boundary; this is our approximation of the Cannon-Thurston map: a space-filling curve. Using veering triangulations we give approximations that fill space more evenly than Thurston's algorithm, which suffers from the problem of "thin necks". The space-filling curves of Peano and Hilbert (and many others) are artificially constructed. Those of Cannon and Thurston arise naturally in the study of hyperbolic three-manifolds. Each curve is canonically associated to a surface contained in a three-manifold. The curve and the space that it fills come from the boundaries at infinity of the universal covers of the surface and manifold.