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

Author: Mladen Bestvina
Journal: Bull. Amer. Math. Soc. 51 (2014), 53-70
MSC (2010): Primary 57M50, 57N10
Published electronically: September 30, 2013
MathSciNet review: 3119822
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Abstract | References | Similar Articles | Additional Information


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
Dedicated: Dedicated to Bill Thurston (1946–2012), who taught us how to think about mathematics
Article copyright: © Copyright 2013 American Mathematical Society