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Category Archives: Hyperbolic geometry
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
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 CAT(K), comparison geometry, convexity, Jacobi fields, nonpositive curvature, Riemannian geometry
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 KahnMarkovic, Wise, HaglundWise and BergeronWise, the proof reduces to showing the following: Theorem … Continue reading
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 3manifold topology (hat tip to Henry … Continue reading
Filling geodesics and hyperbolic complements
Patrick Foulon and Boris Hasselblatt recently posted a preprint entitled “Nonalgebraic contact Anosov flows on 3manifolds”. These are flows which are at the same time Anosov (i.e. the tangent bundle splits in a flowinvariant way into stable, unstable and flow directions) and contact … Continue reading
Quasigeodesic flows on hyperbolic 3manifolds
My student Steven Frankel has just posted his paper Quasigeodesic flows and Mobiuslike 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
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 3manifolds, Groups, Hyperbolic geometry, Uncategorized
2 Comments