Property (T) is a central rigidity property in analytic group theory.
It can be formulated in many equivalent ways, for instance
as a spectral gap property or in terms of group cohomology.
Ozawa showed a remarkable characterization: Property (T) can be detected
by an equation in the group algebra involving sums of squares.
This was fundamental in proving that Aut(F_n) has
Property (T) for n at least 4 via computer assisted methods.
Recently, there has been increasing interest in higher versions of property (T).
Using group cohomology, there are at least
two possible non-equivalent definitions: ona via vanishing and
one via reducedness of cohomology. It is thus natural to explore
whether these higher analogues can be characterized
in terms of equations à la Ozawa. In this series of lectures we want
to discuss (not in 1-to-1 correspondence with lectures):
- Basic definitions of Property (T) and group cohomology in degree 1
- The GNS construction for functions of conditionally negative type and the Delorme-Guichardet theorem
- *-algebras and Ozawa's theorem
- More group cohomology and finiteness properties
- Higher (T) as vanishing of cohomology: the Bader-Nowak characterization
- (joint work with Piotr Nowak) Higher (T) as reducedness of cohomology and the higher GNS construction.