Kazhdan’s Property (T) via Semidefinite Optimization

Following an idea of Ozawa, we give a new proof of Kazhdan’s property (T) for by showing that Δ2 − Δ/6 is a Hermitian sum of squares in the group algebra, where Δ is the unnormalized Laplace operator with respect to the natural generating set. This corresponds to a spectral gap of 1/72 ≈ 0.014 for the associated random-walk operator. The sum-of-squares representation was found numerically by a semidefinite programming algorithm and then turned into an exact symbolic representation, provided in an attached Mathematica file.