# Better pictures of Cannon-Thurston maps

Date started: May 2018
Leads: David Bachman, Saul Schleimer, Henry Segerman

### Abstract

The Cannon-Thurston map is a geometric way to produce sphere-filling curves. Roughly speaking, one starts with a surface $S$ immersed in a three-manifold $M.$ The former has a “circle at infinity” $S^1_\infty$ while the later has a “sphere at infinity” $S^2_\infty$. Thus, sometimes, the immersion $i :S \to M$ leads to a map $i_*$ between the associated objects at infinity. In the good case, when $S$ is a fiber of a fibration of $M$ and when $M$ is hyperbolic, one way to draw the image of $S^1_\infty$ 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 $S$ of the figure-eight knot included into its complement $M$. We seek to understand how this picture was drawn, to draw similar pictures for other pairs $(M, S)$, and to find other techniques for drawing Cannon-Thurston maps.