Algebraic positivity and property (T)

Property (T) for a group G is originally defined in terms of all unitary representations on Hilbert spaces. I will show how to use Positivstellensätze in group rings to reduce property (T) to a specific algebraic question RG in the group ring. Then I will define grading of group by root system and show how it may be used to prove property (T) for whole families of groups with compatible gradings by solving a single equation in the group ring. A specific example of such application is the result that Aut(Fₙ) has property (T) for n ≥ 6.