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

Abstract

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.