Kazhdan's Property (T) via Semidefinite Optimization

Abstract

Following an idea of Ozawa, we give a new proof of Kazhdan's property (T) for $\mathrm{SL}(3,\mathbb{Z})$, by showing that ${\Delta }^2-\frac{1}{6}\Delta$ is a hermitian sum of squares in the group algebra, where $\Delta$ is the unnormalized Laplace operator with respect to the natural generating set. This corresponds to a spectral gap of $\frac{1}{72}\sim 0.014$ 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.

arXiv commentone mathematica notebook and two data files attached

References in corpus (1)

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

Cited by in corpus (1)

• Computing Kazhdan constants by semidefinite programming