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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
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
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
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 central limit theorem, local limit theorem, Markov chain, tiling, train tracks
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
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 Kahn, Markovic, pair of pants, surface groups
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 Bowen, geodesic flow, Hall's marriage theorem, Kahn, LERF, Markovic, pair of pants, surface groups, Waldhausen
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
Geometry and the imagination 