Geometric Analysis and Topology Seminar

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Cube Complexes and the Proof of Thurston's Conjectures

Speaker: Dani Wise, McGill

Location: Warren Weaver Hall 1302

Date: Monday, May 5, 2014, 11 a.m.

Synopsis:

In the 1970's, Thurston conjectured that every closed aspherical 3-manifold has a finite cover that has positive first Betti number, and that every closed hyperbolic 3-manifold has a finite cover that fibers over the circle. Surprisingly, the key to understanding these issues turned out to be the nonpositively curved cube complexes popularized by Gromov. Over the past 20 years, a program was developed to understand groups using cube complexes, incorporating a construction of Sageev and a connection of Haglund-Wise to Coxeter groups. The recent resolution of the surface subgroup problem by Kahn-Markovic supplied the important ingredient to make this program applicable to closed hyperbolic 3-manifolds, and after recent progress of the speaker, the exciting final ingredient was recently completed by Agol. I'll survey these developments without assuming extensive background from the audience.