Ergodic theory and measured group theory, winter semester 2010/2011

--Schedule -- Where and When -- Generalities -- Maths -- Disclaimer --

If there is no opposition, during the semester break the current speaker can choose that he wants to speak also on thursday (14:15, Sitzungszimmer). This will be decided during the standard tuesday meetings.

EMERGENCY: The meeting on thursday, 24.03, is cancelled.

Next meeting: tuesday, 29.03, Sitzungszimmer, 16:15. It will be the last meeting of the seminar.

Schedule

Upcoming lectures

Dates and order can change.

Vadim Alekseev and/or David Kyed, 29.03.2011 Based on a paper by G. Elek and E. Szabo There are two results: the free product of two sofic gorups amalgamated over an amenable group is sofic, and sofic representation of an amenable group is essentially unique. Maybe one can also compare with the paper of Liviu Paunescu

Propositions of other topics

Get in touch if you're interested in preparing one of them.

Based on a paper by Miklos Abert, Andrei Jaikin-Zapirain and Nikolay Nikolov on rank gradient. Among others, interesting results on rank gradient of amenable groups which stress the difference between finaly generated and finitely presented case.

A paper of R. Sauer and A. Thom on a spectral sequence computing l2-Betti numbers.

Theorem of J.Peterson and A.Thom saying that for a torsion-free group with a positive first l2-Betti number, a weak version of the Atiyah conjecture implies existence of a free subgroup. Based on their paper. Compare remarks at the end of an article of I.Epstein and N. Monod on random forests and Dixmier problem.

Recent work of Osin, and Lueck-Osin on possible counterexamples to cost vs first l^2-Betti number and fixed price conjectures: http://arxiv.org/abs/1011.4739, http://arxiv.org/abs/0905.1322

Past lectures

Lukasz Grabowski, 26.10, 9.11, a bit of 16.11 Betti numbers, l^2-Betti numbers, measurable relations. Basic definitions + Lueck's approximation theorem
Rohan Lean, 16.11, 23.11 Based on D. Gaboriau, Invariants l2 de relations d'equivalence et de groupes; l^2-Betti numbers of a relation, proof that they are the same as those of a group when the relation is generated by a free action of the group. Part I
Thomas Schick, 30.11 Second part of the lecture on the paper of Gaboriau.
Jean-Francois Planchat, 7.12 Introduction to amenable groups
Thomas Schick, 14.12, 21.12 Amenability and soficity. A survey. In particular approximation theorems in the context of sofic approximation (for example results from a paper of Andreas Thom. Maybe showing some particular cases of orbit equivalence of amenable groups? Maybe result on direct finiteness of Elek and Szabo? Maybe determinant conjecture for sofic groups by the same authors?
Vadim Alekseev, 06.01, 11.01, 14.01 Survey on results about cost and fixed price property for groups. In particular the theorem that cost of a relation is equal to cost of any treeing of this relation (note that not every relation has a treeing) - the other french paper of Gaboriau. Maybe some particular cases of the theorem that cost of any amenable group is 1?
Lukasz Grabowski, 18.01 Gaboriau-Lyons theorem saying that every nonamenable group has a free group as a measurable subgroup.
Markus Upmeier, 25.01 Applications of bounded cohomology and Gaboriau-Lyons thoerem to Dixmier unitarisibility problem. Based chiefly on a paper by Monod and Ozawa
Lukasz Grabowski, 1.02 Non-unitarizability of the free group on two generators.
David Kyed, 8.02 Introduction to property (T) groups. In particular it's known that their first l^2-Betti number is 0, and it's not known if their cost is 1.
Lukasz Grabowski, 22.02 Theorem of Abert-Nikolov on cost of the action on the boundary of a coset tree. Based on their paper.
Thomas Schick, 24.02 Survey on 3-manifolds + brief intro to Heegard genus conjecture and its relation with Abert-Nikolv theorem.
Thomas Schick, 1.03 On the Atiyah conjecture for subgroups of GL(n,Z). Based on the article of D. Farkas and P.Linnell.

Jean-Francois Planchat, 8.03 Introduction to property tau. Perhaps somewhat based on a paper of M. Abert and G. Elek. or on this or that paper of M. Lackenby.

Thomas Schick, 17.03.2011 Results of M. Lackenby on Heegard genus. Based on this or that paper of M. Lackenby.

Holger Kammeyer, 22.03.2011 On Novikov-Shubin invariants arising from free groups. Based on a paper of Roman Sauer

Where and When

Tuesdays, 16:15-17:45, Sitzungszimmer

Generalities

The primary target is young people, i.e. Master students and fresh PhD students. In case there are couple of people wanting to prepare one of the lectures, preference will be given to those groups. The topics we will cover will be centered around actions of groups on measure spaces, infinite graphs, measurable graphings, soficity (i.e. roughly possibility of approximating infinite graphs by finite ones), and relations to questions in topology and geometry. If time permits we will see some connections with von Neumann algebras,

The idea is to focus on presenting open problems (which are many and interesting) and recent results.

First two lectures will be presented by me. After the first lecture we will hopefully have some general feel of what is it going to be about, and some volunteers will volunteer to prepare subsequent lectures.

If you want to talk about any issues related to the seminar, especially mathematics, feel free to contact me or Thomas. Also, if you want to ask a mathematical question by email, consider using Stud.ip forum of the seminar instead, since then all the participants can benefit from it. Another great place to ask questions is mathoverflow, although questions about basic definitions shoudn't be asked there.

If something doesn't work on this page, I'm the right person to contact.

Maths

16.11.2010

I did a terrrible job last time introducing l^2-Betti numbers in a chaotic way, possibly making the impression that this is a difficult subject. It's far from being true, if you got this impression please do read lecture notes of B. Eckmann.

Amenability will feature more and more in the lectures. While waiting for the "official" introduction by Thomas Schick, you might want to read chapter 5 from a survey by Nathaniel Brown. In particular theorem 5.8.

18.10.2010

In the first two lectures I will talk about measurable relations, Betti numbers, l^2-Betti numbers, and Lueck's approximation theorem.

One thing which bothers me is the following: we will heavily use measure theory, but I definitely don't want to talk about Borel sets, etc. So it would be good if you have some experience with meassures. Otherwise you'll have to believe me that "measurable" is more-or-less another way of saying "defined withouth the axiom of choice". Perhaps skimming through the chapter 7.4 of "A course in harmonic analysis" is a good idea if you want to get a "feel" for all things measurable.

I will largely assume everyone knows what Betti numbers (i.e. dimensions of homology groups of CW-complexes) are. What is going to be useful later is the "harmonic chains" point of view, which is beautifully described in the first chapter of my favourit paper ever:

The whole paper is a really good read.

One important theorem whose proof I'll sketch is the Lueck's approximation theorem. Although the main idea is very simple the technicalities are substantial. Details can be found in W.Lueck's book L^2-Invariants: Theory and Appications to Geometry and K-Theory, chapter 13.

The main short-term target is to define l^2-Betti numbers of a measurable relation and show that l^2-Betti numbers of a group are the same as those of the relation it generates. This was done in the following paper:

We will most likely follow this paper. For those which are more homological-algebra minded, there are two closely related alternative proofs: Finally here are two surveys. The first is more in spirit of what we're going to do: I also latexed my notes from an open problem session in Oberwolfach. It's rather informal. Things in red are things which I coundn't reliably reconstruct. They can be anything from "not true" to "no sense".

Disclaimer

There are already quite a few links to research papers, and probably there will be even more. This doesn't mean you should try to read them all :-). To quote Grothendieck:
"Since then I've had the chance, in the world of mathematics that bid me welcome, to meet quite a number of people, both among my "elders" and among young people in my general age group, who were much more brilliant, much more "gifted" than I was. I admired the facility with which they picked up, as if at play, new ideas, juggling them as if familiar with them from the cradle -- while for myself I felt clumsy, even oafish, wandering painfully up an arduous track, like a dumb ox faced with an amorphous mountain of things that I had to learn (so I was assured), things I felt incapable of understanding the essentials or following through to the end. Indeed, there was little about me that identified the kind of bright student who wins at prestigious competitions or assimilates, almost by sleight of hand, the most forbidding subjects. In fact, most of these comrades who I gauged to be more brilliant than I have gone on to become distinguished mathematicians. Still, from the perspective of 30 or 35 years, I can state that their imprint upon the mathematics of our time has not been very profound. They've all done things, often beautiful things, in a context that was already set out before them, which they had no inclination to disturb. Without being aware of it, they've remained prisoners of those invisible and despotic circles which delimit the universe of a certain milieu in a given era. To have broken these bounds they would have had to rediscover in themselves that capability which was their birth-right, as it was mine: the capacity to be alone."