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- Kähler manifolds and groups, part 2
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- Liouville illiouminated
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- You can solve the cube – with commutators!
- Chiral subsurface projection, asymmetric metrics and quasimorphisms
- Random groups contain surface subgroups
- wireframe, a tool for drawing surfaces
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Category Archives: Surfaces
Slightly elevated Teichmuller theory
Last week at my invitation, David Dumas spoke in the U Chicago geometry seminar and gave a wonderful introductory talk on the theory of convex real projective structures on surfaces. This is the first step on the road to what is colloquially known … Continue reading
Mapping class groups: the next generation
Nothing stands still except in our memory. – Phillipa Pearce, Tom’s Midnight Garden In mathematics we are always putting new wine in old bottles. No mathematical object, no matter how simple or familiar, does not have some surprises in store. My … 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 quasi-trees (i.e. spaces quasi-isometric to trees). The construction is inspired … Continue reading
Random groups contain surface subgroups
A few weeks ago, Ian Agol, Vlad Markovic, Ursula Hamenstadt and I organized a “hot topics” workshop at MSRI with the title Surface subgroups and cube complexes. The conference was pretty well attended, and (I believe) was a big success; … Continue reading
wireframe, a tool for drawing surfaces
The purpose of this brief blog post is to advertise that I wrote a little piece of software called wireframe which can be used to quickly and easily produce .eps figures of surface for inclusion in papers. The main use is … Continue reading
Surface subgroups of Sapir’s group
Let be the free group on two generators, and let be the endomorphism defined on generators by and . We define Sapir’s group to be the ascending HNN extension This group was studied by Crisp-Sageev-Sapir in the context of their … Continue reading
Posted in Ergodic Theory, Groups, Surfaces
Tagged f-folded surface, fatgraph, HNN extension, hyperbolic group, Sapir's group, Stallings folding, surface subgroup
12 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 CAT(K), comparison geometry, convexity, Jacobi fields, nonpositive curvature, Riemannian geometry
2 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
Geometry and the imagination 