Category Archives: Ergodic Theory

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 , , , , | 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

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 Kahn-Markovic, Wise, Haglund-Wise and Bergeron-Wise, the proof reduces to showing the following: Theorem … Continue reading

Posted in 3-manifolds, Ergodic Theory, Groups, Hyperbolic geometry | Tagged , , , , | 8 Comments

Laying train tracks

This morning I was playing trains with my son Felix. At the moment he is much more interested in laying the tracks than putting the trains on and moving them around, but he doesn’t tend to get concerned about whether … Continue reading

Posted in Ergodic Theory, Euclidean Geometry | Tagged , , , , | 19 Comments

Roth’s theorem

I am in Kyoto right now, attending the twenty-first Nevanlinna colloquium (update: took a while to write this post – now I’m in Sydney for the Clay lectures). Yesterday, Junjiro Noguchi gave a plenary talk on Nevanlinna theory in higher dimensions … Continue reading

Posted in Diophantine approximation, Ergodic Theory | Tagged , , , | 10 Comments

Surface subgroups – more details from Jeremy Kahn

Jeremy Kahn kindly sent me a more detailed overview of his argument with Vlad Markovic, that I blogged earlier about here (also see Jesse Johnson’s blog for other commentary). With his permission, this is reproduced below in its entirety. Editorial … Continue reading

Posted in 3-manifolds, Ergodic Theory, Surfaces | Tagged , , , | 4 Comments

Surface subgroups in hyperbolic 3-manifolds

I just learned from Jesse Johnson’s blog that Vlad Markovic and Jeremy Kahn have announced a proof of the surface subgroup conjecture, that every complete hyperbolic -manifold contains a closed -injective surface. Equivalently, contains a closed surface subgroup. Apparently, Jeremy … Continue reading

Posted in 3-manifolds, Ergodic Theory, Surfaces | Tagged , , , , , , , , | 10 Comments

Combable functions

The purpose of this post is to discuss my recent paper with Koji Fujiwara, which will shortly appear in Ergodic Theory and Dynamical Systems, both for its own sake, and in order to motivate some open questions that I find … Continue reading

Posted in Ergodic Theory | Tagged , , , , , , | 2 Comments