Analysis and Probability seminar

Abstract: For a countable group G, Kazhdan’s property (T) is equivalent to the fact that any isometric (hence necessarily affine) action of G on the Hilbert space has a fixed point. This is considered as a form of a strong “rigidity” for groups, and a strong negation of amenability.

Yehuda Shalom conjectured that Burnside groups have Kazhdan’s property (T). I will present a proof that this is not the case: many Burnside groups can act on Hilbert spaces without fixing a point.

Here, the Burnside group means a finitely generated infinite torsion (that is, every element has a bounded order) group of bounded exponent (there is a universal bound on orders of elements).

I will also present related open question.