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 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.
Media
![](https://i0.wp.com/im.icerm.brown.edu/wp-content/uploads/2019/02/gLLAQbecdfffhhnkqnc_120012_0.0-0.0-0.0-0.1666666667-0.0-0.1666666667-0.0-0.3333333333-0.1666666667-0.0-0.0-0.1666666667_0.999999_-1.00001-1.0_-1.00001-1.0_2000_2000_0.3_cool.png?resize=1091%2C1091&ssl=1)
![](https://i0.wp.com/im.icerm.brown.edu/wp-content/uploads/2019/02/cPcbbbiht_12_0.0-0.5-0.0-0.5_0.999999_-1.00001-1.0_-1.00001-1.0_2000_2000_0.3_warm.png?resize=1091%2C1091&ssl=1)
![](https://i0.wp.com/im.icerm.brown.edu/wp-content/uploads/2019/02/gvLQQcdeffeffffaafa_201102_0.5-0.0-0.0-0.0-0.0-0.0-0.5-0.0-0.0-0.0-0.0-0.0_0.999999_-1.00001-1.0_-1.00001-1.0_2000_2000_0.3_neon.png?resize=1091%2C1091&ssl=1)
![](https://i0.wp.com/im.icerm.brown.edu/wp-content/uploads/2019/02/gLMzQbcdefffhhhhhit_122112_0.0-0.0-0.0-0.0-0.0-0.0-0.0-0.0-0.0-0.0-0.5-0.5_0.999999_-1.00001-1.0_-1.00001-1.0_2000_2000_0.3_green.png?resize=1091%2C1091&ssl=1)
![](https://i0.wp.com/im.icerm.brown.edu/wp-content/uploads/2019/01/shader1.jpg?resize=1091%2C681&ssl=1)
References
- William P. Thurston, Three dimensional manifolds, Kleinian groups and hyperbolic geometry, Bull. Amer. Math. Soc. Volume 6, Number 3, May 1982.