Category Archives: Surfaces

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

Posted in Groups, Hyperbolic geometry, Surfaces | Tagged , , , , , , , , | 3 Comments

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

Posted in Ergodic Theory, Groups, Surfaces | Tagged , , , , | Leave a comment

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

Posted in Surfaces, Visualization | Tagged , | 2 Comments

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 , , , , , , | 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 , , , , , | 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

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

The Hall-Witt identity

The purpose of this blog post is to try to give some insight into the “meaning” of the Hall-Witt identity in group theory. This identity can look quite mysterious in its algebraic form, but there are several ways of describing it geometrically which … Continue reading

Posted in Groups, Lie groups, Surfaces, Visualization | Tagged , , , | 1 Comment