Weekly seminar of AG Laboratory

Mahan Mj (Tata Institute)

Monday, 17:00, June 19, 2017: Cannon-Thurston maps and Kleinian groups (1)

Let M be a closed hyperbolic 3-manifold fibering over the circle with fiber a closed surface S. The inclusion of S
into M lifts to a map between universal covers \tilde{S} and \tilde{M}. In the early 80's Cannon and Thurston
showed that this inclusion extends to a continuous map between their compactifications: namely the 2-disk and the
3-ball. This gives rise to a space-filling (Peano) curve from the circle onto the 2-sphere, equivariant under the
action of the fundamental group of S. This led Thurston to the following questions.
1) Is this a general phenomenon for finitely generated discrete subgroups of the isometry group of hyperbolic
3-space?
2) How does this map behave with respect to sequences of representations?
In the first lecture I shall survey an affirmative answer to Question 1. In the second, I shall give a review of
work (joint in parts with C. Series and K. Ohshika) leading to a resolution of Q. 2.

Wednesday, 17:00, June 21, 2017:  Cannon-Thurston maps and Kleinian groups (2)

Friday, June 23, 2017:  Cannon-Thurston maps in Geometric Group Theory

Let M be a closed hyperbolic 3-manifold fibering over the circle with fiber a closed surface S. The inclusion of S
into M lifts to a map between universal covers \tilde{S} and \tilde{M}. In the early 80's Cannon and Thurston
showed that this inclusion extends to a continuous map between their compactifications: namely the 2-disk and the
3-ball. This can be extended to a considerably broader framework in the context of (Gromov) hyperbolic groups.
I shall survey some of the developments in this broader context.