Narutaka Ozawa

Noncommutative real algebraic geometry of Kazhdan's property (T)

I will start with a gentle introduction to the emerging subject of "noncommutative real algebraic geometry," a subject which deals with equations and inequalities in noncommutative algebra over the reals, with the help of analytic tools such as representation theory and operator algebras. I will then present a surprisingly simple proof that a group $G$ has Kazhdan's property (T) if and only if a certain inequality in the group algebra ${\bf R}[G]$ is satisfied. Very recently, Netzer and Thom used a computer to verify this inequality for ${\rm SL}(3,{\bf Z})$, thus giving a new proof of property (T) for ${\rm SL}(3,{\bf Z})$ with a much better estimate of the Kazhdan constant than the previously known.