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

Posted in Complex analysis, Geometric structures, Projective geometry, Surfaces | Tagged , , , , , | Leave a comment

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

Posted in Dynamics, Groups, Hyperbolic geometry, Surfaces | Tagged , , , , , , , | 7 Comments

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 , , , , , , , , | 4 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 , , , , | 1 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