Thurston’s Vision and the Virtual Fibering Theorem for 3-Manifolds

@article{Friedl2014ThurstonsVA,
  title={Thurston’s Vision and the Virtual Fibering Theorem for 3-Manifolds},
  author={Stefan Friedl},
  journal={Jahresbericht der Deutschen Mathematiker-Vereinigung},
  year={2014},
  volume={116},
  pages={223-241}
}
  • Stefan Friedl
  • Published 24 September 2014
  • Mathematics
  • Jahresbericht der Deutschen Mathematiker-Vereinigung
The vision and results of William Thurston (1946–2012) have shaped the theory of 3-dimensional manifolds for the last four decades. The high point was Perelman’s proof of Thurston’s Geometrization Conjecture which reduced 3-manifold topology for the most part to the study of hyperbolic 3-manifolds. In 1982 Thurston gave a list of 24 questions and challenges on hyperbolic 3-manifolds. The most daring one came to be known as the Virtual Fibering Conjecture. We will give some background for the… 
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Background Citations
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4 Citations

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References

SHOWING 1-10 OF 64 REFERENCES
Geometric group theory and 3-manifolds hand in hand: the fulfillment of Thurston’s vision
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.…
Finite covering spaces of 3-manifolds
Following Perelman's solution to the Geometrisation Conjecture, a 'generic' closed 3-manifold is known to admit a hyperbolic structure. However, our understand- ing of closed hyperbolic 3-manifolds…
Three dimensional manifolds, Kleinian groups and hyperbolic geometry
1. A conjectural picture of 3-manifolds. A major thrust of mathematics in the late 19th century, in which Poincare had a large role, was the uniformization theory for Riemann surfaces: that every…
Criterion for Cannon’s Conjecture
The Cannon Conjecture from the geometric group theory asserts that a word hyperbolic group that acts effectively on its boundary, and whose boundary is homeomorphic to the 2-sphere, is isomorphic to…
Virtual Homological Torsion of Closed Hyperbolic 3-manifolds
In this paper, we will use Kahn-Markovic's almost totally geodesic surfaces to construct certain $\pi_1$-injective 2-complexes in closed hyperbolic 3-manifolds. Such 2-complexes are locally almost…
The virtual Haken conjecture
We prove that cubulated hyperbolic groups are virtually special. The proof relies on results of Haglund and Wise which also imply that they are linear groups, and quasi-convex subgroups are…
The topology of four-dimensional manifolds
0. Introduction Manifold topology enjoyed a golden age in the late 1950's and 1960's. Of the mysteries still remaining after that period of great success the most compelling seemed to lie in…
A boundary criterion for cubulation
We give a criterion in terms of the boundary for the existence of a proper cocompact action of a word-hyperbolic group on a ${\rm CAT}(0)$ cube complex. We describe applications towards lattices and…
From Riches to Raags: 3-Manifolds, Right-Angled Artin Groups, and Cubical Geometry
This book presents an introduction to the geometric group theory associated with nonpositively curved cube complexes. It advocates the use of cube complexes to understand the fundamental groups of…
The entropy formula for the Ricci flow and its geometric applications
We present a monotonic expression for the Ricci flow, valid in all dimensions and without curvature assumptions. It is interpreted as an entropy for a certain canonical ensemble. Several geometric…
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