## Geometric group theory and 3-manifolds hand in hand: the fulfillment of Thurston’s vision

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**51**(2014), 53-70 Request permission

## Abstract:

In the late 1970s, Thurston revolutionized our understanding of 3-manifolds. He stated a far-reaching geometrization conjecture and proved it for a large class of manifolds, called Haken manifolds. He also posed 24 open problems, describing his vision of the structure of 3-manifolds.

Pieces of Thurston’s vision have been confirmed in the subsequent years. In the meantime, Dani Wise developed a sophisticated program to study cube complexes and, in particular, to promote immersions to embeddings in a finite cover. Ian Agol completed Wise’s program and, as a result, essentially all problems on Thurston’s list are now solved. In these notes I will outline a proof that closed hyperbolic 3-manifolds are virtually Haken.

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## Additional Information

**Mladen Bestvina**- Affiliation: Department of Mathematics, University of Utah, Salt Lake City, UT 84103
- MR Author ID: 36095
- Received by editor(s): May 22, 2013
- Published electronically: September 30, 2013
- © Copyright 2013 American Mathematical Society
- Journal: Bull. Amer. Math. Soc.
**51**(2014), 53-70 - MSC (2010): Primary 57M50, 57N10
- DOI: https://doi.org/10.1090/S0273-0979-2013-01434-4
- MathSciNet review: 3119822

Dedicated: Dedicated to Bill Thurston (1946–2012), who taught us how to think about mathematics