The Cannon-Thurston map is a geometric way to produce sphere-filling curves. Roughly speaking, one starts with a surface immersed in a three-manifold The former has a “circle at infinity” while the later has a “sphere at infinity” . Thus, sometimes, the immersion leads to a map between the associated objects at infinity. In the good case, when is a fiber of a fibration of and when is hyperbolic, one way to draw the image of would be to draw a black sphere — after all, the curve is sphere-filling!
Thurston’s classic figure (above right, from this paper) shows an approximation of the sphere-filling curve coming from the inclusion of the Seifert surface of the figure-eight knot included into its complement . We seek to understand how this picture was drawn, to draw similar pictures for other pairs , and to find other techniques for drawing Cannon-Thurston maps.
- William P. Thurston, Three dimensional manifolds, Kleinian groups and hyperbolic geometry, Bull. Amer. Math. Soc. Volume 6, Number 3, May 1982.