Working seminar on analytic group theory

(2011-2012)

 

Next talk:

April Tuesday 24th, 13:00

(organized by Dmitry Jakobson and Mikael Pichot)

On a G-dimension for l^p spaces

Location: The seminar meets on Tuesdays at 1pm in BURN1205

Abstract: In this talk, I will shortly describe the von Neumann dimension (or G-dimension) for a discrete and finitely generated group G. In a particular case, this is a (positive) real number which can be associated to closed G-invariant subspaces of l^2(G) which behaves like one expects a dimension to behave. When the group is amenable, it is possible to define such a dimension from a more naive point of view, in particular, one which does not hinge on the Hilbertian structure. This allows to define a "dimension" for subspaces of other classical Banach spaces (e.g. l^p(G) ). I will try to explain why some of the properties still hold while other can no longer hold and give an application to a question of Gaboriau. If time allows, I will attempt a short description of recent work of B.Hayes which extends this to the class of sofic groups, (where even less properties may or are known to hold).

Antoine Gournay

(Partial list of) References:

A. Gamburd, D. Jakobson and P. Sarnak. Spectra of elements in the group ring of SU(2), Jour. of Eur. Math. Soc. 1(1) (1999), 51-85.

See: http://www.math.mcgill.ca/jakobson/papers/fingen.ps

P. Sarnak. "Some applications of modular forms", Chapter 2.

Previous talks:

February Tuesday 14th, 13:00

Spectra of elements in the group ring of SU(2)

Abstract: We present the results from the 1999 paper with A. Gamburd and P. Sarnak "Spectra of elements in the group ring of SU(2)."


We present an elementary method for constructing finitely generated subgroups of SU(2) with the "spectral gap" property. This provides an elementary solution of the Ruziewicz problem on S^2. We also discuss a natural generalization of the Diophantine property for non-commutative subgroups of SU(2) that turned out to be a crucial ingredient in subsequent generalizations of that work by A. Gamburd, J. Bourgain and P. Sarnak.

Dmitry Jakobson

February Tuesday 28th, 13:00

Spectra of elements in the group ring of SU(2) - Part 2

Abstract: We present the results from the 1999 paper with A. Gamburd and P. Sarnak "Spectra of elements in the group ring of SU(2)."


Dmitry Jakobson

S. Hoory, N. Linial and A. Wigderson. Expander graphs and their applications.

See: http://www.cs.huji.ac.il/~nati/PAPERS/expander_survey.pdf

March Tuesday 27th, 13:00

Groups, Zigzags and Expanders

Abstract: We introduce a class of highly connected but sparse graphs called expanders with interesting combinatorial, geometric and algebraic properties. After an informal survey of their pervasive utility throughout mathematics and theoretical computer science, we emphasize two explicit constructions of infinite families of expanders. First, following the original construction of Margulis, we obtain some families as the Cayley graphs of quotients of infinite groups with property (T). Finally, defining the zig-zag product for graphs we obtain a construction of such families related to the semidirect product of certain groups.

Maxime Bergeron