Category Archives: Groups

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

Groups quasi-isometric to planes

I was saddened to hear the news that Geoff Mess recently passed away, just a few days short of his 54th birthday. I first met Geoff as a beginning graduate student at Berkeley, in 1995; in fact, I believe he gave the … Continue reading

Posted in 3-manifolds, Complex analysis, Groups, Hyperbolic geometry, Uncategorized | Tagged , , , , , | 3 Comments

3-manifolds everywhere

When I started in graduate school, I was very interested in 3-manifolds, especially Thurston’s geometrization conjecture. Somehow in dimension 3, there is a marvelous marriage of flexibility and rigidity: generic 3-manifolds are flexible enough to admit hyperbolic structures — i.e. Riemannian metrics … Continue reading

Posted in 3-manifolds, Groups, Hyperbolic geometry | Tagged , , , , | 9 Comments

kleinian, a tool for visualizing Kleinian groups

It’s been a while since I last blogged; the reason, of course, is that I felt that I couldn’t post anything new before completing my series of posts on Kähler groups; but I wasn’t quite ready to write my last … Continue reading

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

Kähler manifolds and groups, part 2

In this post I hope to start talking in a bit more depth about the global geometry of compact Kähler manifolds and their covers. Basic references for much of this post are the book Fundamental groups of compact Kähler manifolds by … Continue reading

Posted in Algebraic Geometry, Complex analysis, Geometric structures, Groups | Tagged , , , , , , , , | Leave a comment

You can solve the cube – with commutators!

After a couple years of living out of suitcases, we recently sold our house in Pasadena, and bought a new one in Hyde Park. All our junk was shipped to us, and the boxes we didn’t feel like unpacking are … Continue reading

Posted in Groups | Tagged , , | 15 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 , , , , | 2 Comments