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 Random groups contain surface subgroups
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Category Archives: Hyperbolic geometry
kleinian, a tool for visualizing Kleinian groups
It’s been a while since I last blogged; the reason, of course, is that I felt that I couldn’t post anything new before completing my series of posts on Kähler groups; but I wasn’t quite ready to write my last … Continue reading
Chiral subsurface projection, asymmetric metrics and quasimorphisms
Last week I was at Oberwolfach for a meeting on geometric group theory. My friend and collaborator Koji Fujiwara gave a very nice talk about constructing actions of groups on quasitrees (i.e. spaces quasiisometric to trees). The construction is inspired … Continue reading
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
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
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