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

Bestvina, Mladen

doi:10.1090/S0273-0979-2013-01434-4

Geometric group theory and 3-manifolds hand in hand: the fulfillment of Thurston’s vision
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Bull. Amer. Math. Soc. 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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