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

Abstract

It is well-known that a finitely generated group $\Gamma$ has Kazhdan's property (T) if and only if the Laplacian element $\Delta$ in ${\mathbb R}[\Gamma]$ has a spectral gap. In this paper, we prove that this phenomenon is witnessed in ${\mathbb R}[\Gamma]$. Namely, $\Gamma$ has property (T) if and only if there are a constant $\kappa>0$ and a finite sequence $\xi_1,...,\xi_n$ in ${\mathbb R}[\Gamma]$ such that $\Delta^2-\kappa\Delta = \sum_i \xi_i^*\xi_i$. This result suggests the possibility of finding new examples of property (T) groups by solving equations in ${\mathbb R}[\Gamma]$, possibly with an assist of computers.