Category Archives: 3-manifolds

Agol’s Virtual Haken Theorem (part 3): return of the hierarchies

Ian gave his second and third talks this afternoon, completing his (quite detailed) sketch of the proof of the Virtual Haken Theorem. Recall that after work of Kahn-Markovic, Wise, Haglund-Wise and Bergeron-Wise, the proof reduces to showing the following: Theorem … Continue reading

Posted in 3-manifolds, Ergodic Theory, Groups, Hyperbolic geometry | Tagged , , , , | 8 Comments

Agol’s Virtual Haken Theorem (part 1)

I am in Paris attending a workshop at the IHP where Ian Agol has just given the first of three talks outlining his proof of the Virtual Haken Conjecture and Virtual Fibration Conjecture in 3-manifold topology (hat tip to Henry … Continue reading

Posted in 3-manifolds, Groups, Hyperbolic geometry, Surfaces | Tagged , , , , , , | 4 Comments

Filling geodesics and hyperbolic complements

Patrick Foulon and Boris Hasselblatt recently posted a preprint entitled “Nonalgebraic contact Anosov flows on 3-manifolds”. These are flows which are at the same time Anosov (i.e. the tangent bundle splits in a flow-invariant way into stable, unstable and flow directions) and contact … Continue reading

Posted in 3-manifolds, Dynamics, Hyperbolic geometry | Tagged , , , , , | 4 Comments

Quasigeodesic flows on hyperbolic 3-manifolds

My student Steven Frankel has just posted his paper Quasigeodesic flows and Mobius-like groups on the arXiv. This heartbreaking work of staggering genius interesting paper makes a deep connection between dynamics, hyperbolic geometry, and group theory, and represents the first significant progress … Continue reading

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Rotation numbers and the Jankins-Neumann ziggurat

I’m in Melbourne right now, where I recently attended the Hyamfest and the preceding workshop. There were many excellent talks at both the workshop and the conference (more on that in another post), but one thing that I found very interesting … Continue reading

Posted in 3-manifolds, Dynamics | Tagged , , | 5 Comments

Hyperbolic Geometry Notes #5 – Mostow Rigidity

1. Mostow Rigidity For hyperbolic surfaces, Moduli space is quite large and complicated. However, in three dimensions Moduli space is trivial: Theorem 1 If is a homotopy equivalence of closed hyperbolic manifolds with , then is homotopic to an isometry. … Continue reading

Posted in 3-manifolds, Groups, Hyperbolic geometry, Uncategorized | 2 Comments

Knots with small rational genus

I recently uploaded a paper to the arXiv entitled Knots with small rational genus, joint with Cameron Gordon. The genesis of this paper was a couple of nice (and related) talks at Caltech by Matthew Hedden and Jake Rasmussen in … Continue reading

Posted in 3-manifolds, Hyperbolic geometry, Knot theory | Tagged , , , , , , | 2 Comments