Cannon-Thurston maps

Say you have a genus-2 surface and a diffeomorphism of that surface to itself, then you can make a 3-manifold by crossing that surface with the unit interval to form a “cylinder” and then use the diffeomorphism to glue one end of the cylinder to the other. Thurston showed that if you pick a nice homeomorphism (a pseudo-Anosov map), then you can give your 3-manifold a hyperbolic structure.

As a geometric group theorist, I want to understand the relationship between the fundamental group of the surface and the fundamental group of the hyperbolic 3-manifold. (For those of the representation theory persuasion, this map is a representation of the surface group into PSL(2,C).) In this talk I’ll discuss what Cannon and Thurston discovered if you look at the “behavior at infinity” of these groups. Then I’ll close by telling you a bit about generalizations to this construction that I (and my coauthors) find interesting.