Category Archives: Hyperbolic geometry

Cube complexes, Reidemeister 3, zonohedra and the missing 8th region

There is an old puzzle which starts by asking: what is the next number in the sequence 1,2,4,? We are supposed to recognize the start of the sequence and answer that the next number is surely 8, because the first … Continue reading

Posted in 3-manifolds, Groups, Hyperbolic geometry, Polyhedra | Tagged , , , , | 1 Comment

Thurston talks on geometrization at Harvard

In winter and spring of 2001, Nathan Dunfield and I ran a seminar at Harvard whose purpose was to go through Thurston’s proof of the geometrization theorem for Haken manifolds. This was a very useful and productive exercise, and there … Continue reading

Posted in Hyperbolic geometry, Uncategorized | Tagged , , | Leave a comment

Random turtles in the hyperbolic plane

My eldest daughter Lisa recently brought home a note from her school from her computer class teacher. Apparently, the 5th grade kids have been learning to program in Logo, in the MicroWorlds programming environment. I have very pleasant memories of … Continue reading

Posted in Hyperbolic geometry, Probability, Visualization | Tagged , , , , , , | 5 Comments

Upper curvature bounds and CAT(K)

I am currently teaching a class at the University of Chicago on hyperbolic groups, and I have just introduced the concept of -hyperbolic (geodesic) metric spaces. A geodesic metrix space is -hyperbolic if for any geodesic triangle , and any … Continue reading

Posted in Hyperbolic geometry, Surfaces | Tagged , , , , , | 2 Comments

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