  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.
