Kazhdan's Property (T) via Semidefinite Optimization

TL;DR

This paper introduces a novel proof of Kazhdan's property (T) for SL(3,Z) using semidefinite programming to find a sum of squares representation of a group algebra element, revealing a spectral gap.

Contribution

It provides a new proof of Kazhdan's property (T) for SL(3,Z) via semidefinite optimization, connecting spectral gaps with sum of squares in group algebras.

Findings

Numerical sum of squares representation obtained through semidefinite programming.

Exact symbolic sum of squares representation provided in supplementary material.

Spectral gap of approximately 0.014 established for the associated random walk.

Abstract

Following an idea of Ozawa, we give a new proof of Kazhdan's property (T) for ${\rm SL}(3,\mathbb Z)$, 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.