I’ve recently been looking at the following paper in which -TQFT anomalies are treated carefully and various old constructions of Turaev and Walker are elucidated:
Gilmer, P.M. and Masbaum, G., Maslov Index, Mapping Class Groups, and TQFT, Forum Math. 25 (2013), 1067-1106.
It makes me think a lot about just what the anomaly `actually means’… (more…)
Mark your calendars now: in June 2014, Cornell University will host “What’s Next? The mathematical legacy of Bill Thurston”. It looks like it will be a very exciting event, see the (lightly edited) announcement from the organizers below the fold.
Bill Thurston passed away yesterday at 8pm, succumbing to the cancer that he had been battling for the past two years. I don’t think it’s possible to overstate the revolutionary impact that he had on the study of geometry and topology. Almost everything we blog about here has the imprint of his amazing mathematics. Bill was always very generous with his ideas, and his presence in the community will be horribly missed. Perhaps I will have something more coherent to say later, but for now here are some links to remember him by:
In the previous post, we discussed MIST presentations for mapping class groups of closed oriented surfaces of genus 1 and 2, and (in a weaker sense) also for genus 3. How might we set about obtaining good presentations for mapping class groups of surfaces of higher genus? A-priori, this looks as though it might be a difficult problem.
The breakthrough was a paper by Hatcher and Thurston. The background for their idea was a result of Brown about how to deduce a presentation of the group from a finite description of its action on a simply-connected simplicial complex . The mapping class group acts on a surface rather than on a simply-connected complex; but simply-connected complexes can be built out of a choice of curves on the surface. Hatcher and Thurston make such a choice, and construct the cut-system complex, which they show to be simply-connected using Morse-Cerf theory. This gives an algorithm which in principle constructs a finite presentation for a mapping class group of a surface of arbitrarily high genus. All papers about presentations of mapping class groups for surfaces of arbitrary genus seem to factor through these ideas of Hatcher and Thurston.
One of low dimensional topology’s most popular groups is the mapping class group of a surface. We care about mapping class groups because of how they relate to Heegaard splittings of 3-manifolds. Number theorists and physicists care about mapping class groups because mapping class groups are orbifold fundamental groups of moduli spaces of Riemann surfaces.
Being an unsophisticated sort of a bloke when it comes to group theory, nothing makes me feel that I understand a group like a good concrete presentation of that group, with generators and relations. In my opinion, a good presentation of a group should have the following properties, which I’ll call MIST for fun.
- Memorable- The presentation should be easy to remember.
- Informative- The generators and the relations should be conceptually meaningful and natural, and should tell me something enlightening about the group.
- Simple- The presentation should be easy to work with, by which I mean mainly that it should be short, with as few generators and relations as possible, and with the relations being as short as possible.
- Typical- It should fit into a bigger picture, by which I mean mainly that it should relate easily to similar presentations of similar groups.
Lots of work has been done to find the best presentation for a mapping class group. But I think there’s still a (relatively short) way to go; and I think that, if the work’s worth doing, then it’s worth doing right. I’ve been thinking a bit about presentations of mapping class groups for a summer course I’m teaching, and I wish I could teach a MIST presentation. But not enough is known, and I don’t know enough about what is known, so in this post I’ll briefly summarize my understanding of the state of the art for mapping class group presentations for genus (higher genus next time), and you can tell me where I’m wrong and where there’s more to be known!
Here’s a neat result about mapping class goups of Heegaard splittings that was proved in a recent preprint  by Marion Moore and Matt Rathbun: The mapping class group of a Heegaard splitting is determined by the coarse geometry of the curve complex for the Heegaard surface. In particular, a Heegaard splitting determines two quasi-convex subsets of the complex of curves for the Heegaard surface and one can define the quasi-mapping class group for a Heegaard splitting in terms of the quasi-isometries of the complex that keep each set within a bounded neighborhood of itself. Their result shows that (modulo a technicality in genus two) the quasi-Mapping class group of a Heegaard splitting is isomorphic to its mapping class group. (more…)
During a recent visit, number theorist Jordan Ellenberg told me about a “time-worn analogy” between
(a) A pseudo-Anosov homeomorphism acting on a surface.
(b) The Frobenius automorphism of a smooth algebraic curve .
Jordan has two very interesting posts on this subject, one on what the dilatation should be in case (b) and a recent one where he discusses the finite field analogue of the following question related to the Virtual Haken Conjecture:
Conjecture: A hyperbolic 3-manifold which fibers over the circle has a finite cover with .
As I noted earlier, this is known when the fiber has genus two, or more broadly if the monodromy is hyperelliptic. Intriguingly, Jordan explains the analogous conjecture in the context of (b) is also known in exactly this case…
Here’s a recent preprint that sounds pretty interesting by Behrstock, Drutu and Sapir . The asymptotic cone of a metric space X is a new metric space that one constructs by scaling the metric on X by smaller and smaller numbers (i.e. you define for small s) and taking a limit as the scaling factor goes to zero. (Actually, you take an ultralimit, which is determined by an ultrafilter, which I won’t explain here. But I do plan on trying to use the prefix “ultra” in my own definitions whenever I can.) The asymptotic cone is a popular construction in coarse geometry because when you shrink your metric like this, the coarse features of the space turn into Lipschitz features. Whatever is left in the limit is completely determined by the large scale geomety. For example, the asymptotic cone of a delta-hyperbolic space is always a tree. The asymptotic cone of Euclidean space is Euclidean space. The asymptotic cone of any bounded-diameter space is a point.
Behrstock, Drutu and Sapir look at the asymptotic cone of the mapping class group of a surface. One does this by choosing a finite generating set for this group, then constructing the Cayley graph for the group and the generating set, then setting each edge length equal to one to make the Cayley graph a metric space. The resulting space is somehow very close to being delta-hyperbolic (it’s related to the complex of curves, which is in fact delta hyperbolic) but it it’s not quite delta hyperbolic. It has these large Euclidean subspaces that come, for example, from taking Dehn twists along a collection of disjoint essential simples closed curves in the surface. These Dehn twists commute with each other so the subgroup of the mapping class group is Abelian. In the geometric picture, an Abelian group is Euclidean (since it is of the form ) and triangles in this Euclidean subspace are not delta thin. (Update: See Jason Behrstock’s comment for more on this.) But the philosophy is that if you find a way to ignore this flat regions, the space looks delta-hyperbolic.
Since the geometry of the mapping class group looks delta hyperbolic when you ignore the Euclidean parts, one would expect the asymptotic cone to combine tree-like features with Euclidean features. Behrstock, Drutu and Sapir show that there is a Lipschitz map from the asymptotic cone of the mapping class group into a direct product of trees. So, while the geometry doesn’t necessarily look like such a product, the topology of the asymptotic cone does. The direct product accounts for the large flat regions in the group. while the trees account for the delta-hyperbolic remainder.